Frank van den Bosch

Cosmological Constraints from a Combined Analysis
of Clustering & Galaxy-Galaxy Lensing in the SDSS
Frank van den Bosch
Yale University
In collaboration with:
Marcello Cacciato (Leiden), Surhud More (IPMU),
Houjun Mo (UMass), Xiaohu Yang (SHAO)
Halo Occupation Modelling: Motivation & Goal
Our main goal is to study the Galaxy-Dark Matter connection;
i.e., what galaxy lives in what halo?
To constrain the physics of Galaxy Formation
To constrain cosmological parameters
Four Methods to Constrain Galaxy-Dark Matter Connection:
Frank van den Bosch
Large Scale Structure
Satellite Kinematics
Galaxy-Galaxy Lensing
Abundance Matching
Yale University
The Conditional Luminosity Function
The CLF Φ(L|M ) describes the average number of galaxies
of luminosity L that reside in a halo of mass M.
Φ(L) =
�L�M =
�N �M =
�
�
�
Φ(L|M ) n(M ) dM
Φ(L|M ) L dL
Φ(L|M ) dL
Describes occupation statistics of dark matter haloes
Links galaxy luminosity function to halo mass function
Holds information on average relation between light and mass
see Yang, Mo & vdBosch 2003
Frank van den Bosch
Yale University
The CLF Model
We split the CLF in a central and a satellite term:
Φ(L|M ) = Φc (L|M ) + Φs (L|M )
For centrals we adopt a log-normal distribution:
� �
�2 �
1
ln(L/Lc )
dL
√
√
Φc (L|M )dL =
exp −
L
2πσc
2σc
For satellites we adopt a modified Schechter function:
φs
Φs (L|M )dL =
Ls
�
L
Ls
�αs
�
�
2
exp −(L/Ls ) dL
Note: {Lc , Ls , σc , φs , αs } all depend on halo mass
Free parameters are constrained by fitting data.
Frank van den Bosch
Yale University
CLF Constraints from Group Catalogue
Yang, Mo & vdB (2008)
Frank van den Bosch
Yale University
Clustering of Galaxy Groups
Clustering of groups directly probes clustering of Dark Matter Haloes.
Hierarchical Prediction: More massive haloes are more strongly clustered
Galaxy-Group Cross Correlation confirms prediction!
Results independent of halo-mass indicator.
Wang et al., 2008, ApJ, 687, 919
Frank van den Bosch
Yale University
Mass Dependence of Halo Bias
WMAP3 cosmology
WMAP1 cosmology
Inferred Halo Bias in good agreement with
predictions for concordance cosmology
Wang et al., 2008, ApJ, 687, 919
Frank van den Bosch
Yale University
Luminosity & Correlation Functions
DATA: more luminous galaxies are more strongly clustered
LCDM: more massive halos are more strongly clustered
CONCLUSION: more luminous galaxies reside in more massive halos
Frank van den Bosch
Yale University
Cosmology Dependence
Cacciato et al. (2009)
Frank van den Bosch
Yale University
Galaxy-Galaxy Lensing
The mass associated with galaxies lenses background galaxies
background sources
lensing due to foreground galaxy
Lensing causes correlated ellipticities, the tangential shear, γt, which
is related to the excess surface density, ∆Σ , according to
γt (R)Σcrit = ∆Σ(R) = Σ̄(< R) − Σ(R)
∆Σ is line-of-sight projection of galaxy-matter cross correlation
� Ds
Σ(R) = ρ̄
[1 + ξg,dm (r)] dχ
0
Frank van den Bosch
Yale University
Galaxy-Galaxy Lensing: The Data
Number of background sources per lens is limited
Measuring shear with sufficient S/N requires stacking of many lenses
∆Σ(R|L1 , L2 ) has been measured using the SDSS by
Mandelbaum et al. (2006), using different bins in lens-luminosity
Mandelbaum et al. (2006)
Frank van den Bosch
Yale University
How to interpret the signal?
Stacking
Because of stacking the lensing signal is difficult to interpret
In order to model the data, what is required is:
Pcen (M |L)
Psat (M |L)
fsat (L)
These can all be computed from the CLF...
For a given Φ(L|M ) we can predict the lensing signal ∆Σ(R|L1 , L2 )
Frank van den Bosch
Yale University
Constraining Cosmology
Our Version of the Halo Model
P
P
1h
2h
1
(k) = 2
ρ̄
�
1
(k) = 2
ρ̄
�
R2
R1
R
dM M 2 n(M ) |ũ(k|M )|2
dM1 M1 n(M1 ) ũ(k|M1 )
�
halo exclusion
�
dM2 M2 n(M2 )ũ(k|M2 ) Q(k|M1 , M2 )
∞
sin kr 2
Here Q(k|M1 , M2 ) = 4π
[1 + ξhh (r|M1 , M2 )]
r dr
kr
rmin
with 1 + ξhh (r|M1 , M2 ) = [1 + bh (M1 )bh (M2 )ζ(r)ξmm (r)] Θ(r − rmin )
describes the fact that dark matter haloes are clustered, as described by
the halo-halo correlation function, ξhh (r|M1 , M2 ) , and takes halo exclusion
into account by having rmin = R1 + R2
Smith, Scoccimarro & Sheth (2007)
Smith, Desjacques & Marian (2011)
NOTE: ζ(r) = radial bias function (Tinker et al. 2005)
ξmm (r) = non-linear matter auto-correlation function (Smith et al. 2003)
Frank van den Bosch
Yale University
Towards a faster halo model...
P
2h
1
(k) = 2
ρ̄
�
dM1 M1 n(M1 ) ũ(k|M1 )
�
dM2 M2 n(M2 )ũ(k|M2 ) Q(k|M1 , M2 )
No Halo Exclusion (speed-up: ~10x)
� �
�2 �
1
sin kr 2
2h
P (k) = 4π
dM M n(M ) bh (M ) ũ(k|M )
ζ(r) ξmm (r)
r dr
ρ̄
kr
Use Linear Matter Power Spectrum
� �
�2
1
2h
P (k) =
dM M n(M ) bh (M ) ũ(k|M ) Plin (k)
ρ̄
Frank van den Bosch
Yale University
...but at the loss of accuracy
Frank van den Bosch
Yale University
The Galaxy-Galaxy Correlation Function
P
P
1h
2h
1
(k) = 2
ρ̄
1
(k) = 2
ρ̄
�
�
dM M 2 n(M ) |ũ(k|M )|2
dM1 M1 n(M1 ) ũ(k|M1 )
�
dM2 M2 n(M2 )ũ(k|M2 ) Q(k|M1 , M2 )
The above equations describe the non-linear matter power-spectrum.
It is straightforward to use same formalism to compute power spectrum of galaxies:
Simply replace
M
�N �M
→
ρ̄m
n̄g
ũ(k|M ) → ũg (k|M )
where �N �M describes the average number of galaxies (with certain properties)
in a halo of mass M . Thus, the halo model combined with a model for the halo
occupation statistics, allows a computation of ξgg (r)
Frank van den Bosch
Yale University
Comparison with Mock Catalogues
Run numerical simulation of
structure formation (DM only)
Identify DM haloes, and
populate them with galaxies
using a model for the CLF.
Compute galaxy-galaxy
correlation functions for
various luminosity bins.
Use analytical model to
compute the same, using the
same model for the CLF.
Our model is accurate
to better than ~5%
van den Bosch et al. (2013)
Frank van den Bosch
Yale University
Residual Redshift Space Distortions
To avoid redshift space
distortions, one typically uses
projected correlation function
Because of limitations of data,
one can only integrate out to
finite radius, rmax
wp = 2
�∞
ξgg (rp , rπ ) drπ = 2
rp
0
wp = 2
r�max
0
�∞
√
ξgg (rp , rπ ) drπ �= 2
r dr
ξgg (r) �
r2 − rp2
2 +r 2
rp
max
�
rp
ξgg (r) �
r dr
r2 − rp2
The resulting, residual z-space distortions easily exceed 20% at rp ~20 Mpc/h
(Padmanabhan+07; Norberg+09; Baldauf+10)
We correct for these
residual redshift space
distortions using modified
Kaiser formalism. This is
accurate to better
than 2%.
Frank van den Bosch
Yale University
Fiducial Model
Total of 16 free parameters:
- 9 parameters to describe CLF
- 5 cosmological parameters; Ωm , Ωb , σ8 , ns , h
- 2 nuisance parameters; ψ , η
Total of 176 data points.
WMAP7 priors on Ωb , ns , h
Correction for residual redshift space distortions
Dark matter haloes follow NFW profile
Radial number density distribution of satellites
follows that of dark matter particles.
Halo mass function, halo bias function and radial
bias function fromTinker et al. (2005, 2009, 2010).
Frank van den Bosch
Yale University
Results: Clustering Data
SDSS: Zehavi et al (2011)
Cacciato et al. (2013)
Frank van den Bosch
Yale University
Results: Lensing Data
SDSS: Mandelbaum et al (2006)
Cacciato et al. (2013)
Frank van den Bosch
Yale University
Cosmological Constraints
Cacciato et al. (2013)
Frank van den Bosch
Yale University
and then there was Planck...
Mandelbaum+13
(68% CL)
Frank van den Bosch
Yale University
Cosmological Constraints from Peculiar Velocities
Planck
Hudson & Tumbull (2012)
Frank van den Bosch
Yale University
Cosmological Constraints
?
B
M
C
k
c
n
a
l
P
.
w
g
n
o
r
w
s
i
t
a
h
w
Hudson+Tumbull 2012
(68% CL)
Frank van den Bosch
Yale University
Degeneracies
usat(r|M) different from uDM(r|M)
Residual Redshift Space Distortions
non-Poissonian P(Nsat|M)
no Sloan Great Wall
Cacciato et al. (2013)
Frank van den Bosch
Yale University
Conclusions
Recent years have seen enormous progress in establishing
the galaxy-dark matter connection, including its scatter!
Different methods (group catalogues, satellite kinematics,
galaxy-galaxy lensing, clustering & abundance matching) now
all yield results in good mutual agreement.
Combination of galaxy clustering and galaxy-galaxy lensing
can constrain cosmological parameters.
This method is complementary to and competitive with BAO,
cosmic shear, SNIa & cluster abundances.
Results in excellent agreement with CMB constraints from
WMAP7 with similar analysis by Mandelbaum+12, and with
recent peculiar velocity analysis by Hudson &Tumbull 2012...
[but tension w. Planck CMB]
Frank van den Bosch
Yale University
Halo Occupation Statistics
Cacciato et al. (2013)
Posterior on CLF perfectly consistent with results from
Galaxy Group Catalogues by Yang, Mo & vdB (2008).
Frank van den Bosch
Yale University
Covariance Matrix
Covariance matrix has block
diagonal form.
cosmological
parameters
Little correlation between
cosmological parameters,
and other parameters.
nuisance
params
Nuisance parameters are
mainly correlated with the
satellite CLF parameters
central CLF
parameters
Our results are robust to
our particular parameterization
of the CLF.
satellite
CLF
parameters
More et al. (2013)
Frank van den Bosch
Yale University
Cosmological Constraints
Cacciato et al. (2013)
Frank van den Bosch
Yale University
CLF parameters
Frank van den Bosch
Yale University
Nuisance Parameters
Cacciato et al. (2013)
c(M, z) = (1 + η) c̄(M, z)
Frank van den Bosch
Non-linear (radial) halo bias
Yale University
Title
Frank van den Bosch
Yale University
Galaxy Clustering: The Data
different luminosity bins
Wang et al. (2007)
More luminous galaxies are more strongly clustered
Frank van den Bosch
Yale University
Occupation Statistics from Clustering
Galaxies occupy dark matter halos
CDM: more massive halos are more strongly clustered
Clustering strength of given population of galaxies
indicates the characteristic halo mass
Clustering strength measured by correlation length r0
CAUTION: results depend on cosmology
Frank van den Bosch
Yale University
Results from MCMC Analysis
Model fits data extremely well withχ2red � 1
Same model in excellent agreement with results
from SDSS galaxy group catalogue
Frank van den Bosch
Cacciato, vdB et al. (2009)
Yale University
Galaxy-Galaxy Lensing: Results
Cacciato et al. (2009)
Combination
NOTE: this
ofisclustering
not a fit, &but
lensing
a prediction
can constrain
based cosmology!!!
on CLF
Frank van den Bosch
Yale University
Fisher-Forecasting
More et al. (2013)
Frank van den Bosch
Yale University
Neutrino Mass
Frank van den Bosch
Yale University
Dark Energy
Frank van den Bosch
Yale University
Dark Energy
Frank van den Bosch
Yale University
Comparison with other methods
Frank van den Bosch
Yale University