1. MULTIPROBES DETERMINATION OF COSMOLOGICAL PARAMETERS AND CONSISTENCY CHECK. A. Tilquin (CPPM)
Main people involved: Charling TAO , ZHANG XinMin, André TILQUIN , QIAN Zuxuan , ZHAO Gongbo , XIA Junqing , FAN Zuhui , SUN Lei, Jean Marc VIREY , LI Hong , Zhu Zhonghong , Christian MARINONI , Jean Paul KNEIB, Stephanie JOUVEL, QIN Bo . DENG Jinsong , CHAO Wu , ZHOU Xu , WANG Xiaofeng
CPPM/CPT/IHEP/LAM/PKU/NAOC/Tsinghua U/Beijing Normal U. + collaborators

2. Outlook Introduction Probes used Statistic in ZEN
Full combination results
Cross check and systematic
Summary

3. Cosmology group in FCPPL
Main topic is cosmology (dark energy/ dark matter)
Phenomenology
Data analysis : combinations and systematic studies
7 labs:
5 in Beijing: IHEP+PKU+NAOC+Tsinghua U.+Beijing Normal U.
2 in Marseille: CPPM+CPT
One join thesis between CPPM and PKU: SUN Lei
Some long term missions every year:
Sun Lei, Pr QIN Bo, DENG Jinsong, Charling Tao, Zuxuan Qian and myself
Common publications:
-“Galaxy Distribution as a Probe of the Ringlike Dark Matter Structure in the Galaxy Cluster CL ” Bo Qin, Huan-Uan Shan,Andre Tilquin
-“Catastrophic Photo-z Errors and the Dark Energy Parameter Estimates with Cosmic Shear” Lei Sun, Zu-Hui-Fan, Charling Tao, J-P Kneib, Stephanie Jouvel, Andre Tilquin
-“Probing for Dynamics of Dark Energy and Curvature of Universe with Latest Cosmological Observations” Gong-Bo Zhao, Jun-Qing Xia, Hong Li, Charling Tao, Jen-Marc Virey, Zong-Hong Zhu, Ximmin Zhang
-“Constraining Cosmological Parameters with Observational Data Including Weak Lensing Effects” Hong Li, Jie Liu, Jun-Qing Xia, Zu-Hui Fan, Charling Tao, Andre Tilquin, Xinmin Zhang

4. Dark energy from observations
SNe Ia
CMB(WMAP)
Weak lensing
LSS
Universe is mainly dark
~70% DE and ~25% DM

5. What is dark energy? g + G = T Modified Gravity/GR ?
Non minimal Couplings?
Extra-Dimensions?
Anisotropy/ inhomogeneity effects?
Negative energy?
Cosmological Constant.
New form of « field/matter? » Quintessence?
Unified Dark Matter?
New constant of nature
Acts like an anti-gravitational force :
F-r
w=p/r =w(z)
LCDM w=p/r= cte=-1
w=p/r =/=cte= w(z) effectif
How can we distinguish?

6. Combinations!
Cosmological data has reached a level where cross correlations can give:
Precise measurement of equation of state
Address new physics like modification of gravity
Precise statistical and systematical study
Combinations of probes help:
To lift degeneracies between cosmological parameters
To cross correlate measurement and perform consistency check ->use 3 probes to predict a 4th one
In all the following I will only assume:
A flat Universe
A specific parameterization of equation of state:
w(z) = w0+wa z/(1+z)
Use DE perturbation (CMB).
No prior!

7. Probe I and software. The CMB temperature maps.
WMAP 5 years data taking
Latest version of CMBEasy (c++)

8. Probes II : Supernovae Ia
SN 1a are not standard candle but are standardisable to some level
The strech correction is higher than the dark energy effect we are going to measured ! i.e systematic?

9. Probes II and software Supernovae type Ia: Cosmoshow (c):
SNLS 1 year sample (Astier et al. 2006)
115 SNIa up tp z=1.
Union sample (Kowalski et al. 2008):
307 SNIa up to z = 1.6
Constitution sample (Hicken et al ):
309 SNIa up to z=1.4
Luminosities distance calibrated with SALT2 (Guy et al. 2007)
SN2009 sample (Kessler at al. 2009):
287 SNIa up to z=1.6
Including SDSS sample and tighter selection
Cosmoshow (c):
Based on kosmoshow (A. Tiquin 200x)

10. SNIa samples m 0.43±0.14 0.45±0.08 0.45±0.07 0.18±35(0.06) w0
wa=-7.3±17
m=0.45±0.08
w0=-1.0±0.7
wa=-5.9±10
m=0.45±0.07
w0=-1.0±0.7
wa=-7.6±10
m=0.18±35
w0=-1.0±40
wa=1.0±89
SNLS
UNION
CONSTITUTION
SN2009 m
0.43±0.14
0.45±0.08
0.45±0.07
0.18±35(0.06) w0
-0.7±1.3
-1.0±0.7
-1.0±40(0.8) wa
-7.3±17
-5.9±10
-7.6±10
1.0±87(11)

11. Probes III and software
Baryonic Acoustic Oscillations:
Space correlations of luminous red galaxies:
from SDSS3 (Eisenstien et al. 2005)
BAOfit (F99) (S. Lei 2009):
Based on a set of fitting formulae (Hu and Eisenstein 1997)

12. Probes IV and software Weak lensing surveys (Benjamin et al. 2007):
CFHTLS 57 degree2 (Fu et al. 2009)
Garching-Bonn Deep Survey of 13 degree2 (Hetterscheidt et al. 2007)
Red-Sequence Cluster Survey of 53 degree2 (Hoechstra et al. 2006)
VIRMOS-DESCART of 8.5 d2
(Van Waerbeke et al. 2005)
E-mode
B-mode
WLfit (IDL) (S. Lei 2009):
Based on icosmo
(Refregier et al. 2008)

13. The ZEN program. Based on loglikelihood and frequentist statistics
Full combinations tool:
It can use any new probes
It can manage any languages (c++,idl,F99..)
Intensive use of datagrid (EGEE-Euchina) to speed up processing.
Parallelized calculation of best loglikelihood
4 years of CPU in 2 or 3 days
It’s a public software.

14. The ZEN statistic. ZEN is based on frequentist statistic.
Errors are calculated in 2 different ways
Solving 2 = 2min+s2
But error on Cl’s are not Gaussian
s2 should be calibrated
Using Montecarlo technique
Simulate several hundred experiments
taking into account non-gaussianity.
Error calculated with second order moments
on best fit values:
Ui,j={<ij> - <i> <j>}
MCMC
Li et al. 2009

15. Combination results (Flat Univers)
The most constraining probe is WMAP5: b,h,ns,8,wa
The other 3 probes (SN,BAO,WL) are equivalent
All central values agree perfectly except: b (~ 1)
All combinations compatible with CDM

16. Cross check and systematics
Use the previous result to search from potential deviation from CDM or unknown systematic.
The idea is:
To use the cosmology found by 3 probes to predict the “theoretical” observable of the 4th probe.
To compute the error on this observable induced by the error from the 3 probes prediction.
{in}n=1,200{mn(zj)}2mag = <mn2(z)>-<mn>2
Compare this prediction with the measured observable of the 4th probe.

17. (CMB+BAO+WL)%SN-UNION
1 prediction
Best SN fit
CDM SN best fit: m=0.290.02
CMB+BAO+WL best fit parameters:
mF = 0.280.02; w0F = -0.980.25; waF =0.000.9
We compute the SNIa magnitude in each z bin
m(z) = ms+5log(DL(mF,w0F,waF ))
1 prediction
Best SN fit
CDM SN best fit: m=0.290.02
UNION SAMPLE
1 predicted magnitude
CMB+BAO+WL can constraint SN magnitude at a 8% level up to z=1.7
SN seams brighter at z>1 (1.5 )
Selection bias on HST Supernovae? Or SN evolution?

18. (CMB+BAO+WL)%SN-CONSTITUTION
More nearby and selected sub-sample of the Union SNIa at z>1
Better agreement at z>1
Indicates that problem with Union sample is SN selection (Manquist bias)
CONSTITUTION
1 prediction
Best SN fit
CDM SN best fit: m=0.290.02

19. (CMB+BAO+WL)%SN2009 Use SDSS SN sample at intermediate z
More restricted selection of SNIa at high z
Perfect agreement between CMB+BAO+WL prediction and SNIa
1 prediction
Best SN fit
CDM SN best fit: m=0.290.02

20. SN evolution constraint
To estimate a possible evolution we fit a linear dependency of magnitude with redshift.
The error on difference magnitude between prediction and data is given by :
with I,,j1
SN sample
Magnitude evolution d/dz
SNLS 1st year
-0.010.06
Union
0.0050.07
Constitution
0.0080.07
SN2009
0.010.05
No evolution greater than 8% in luminosity (6% for SN2009) is observed up to z = 1.
Light curve calibration, luminosity galaxy subtraction, dust, or whatever we can think of have a total effect lower than 8% at z=1!

21. How it translates in systematic
Use maximum evolution to compute maximum bias on cosmological parameter.
Fisher analysis
Accurate contour
Bias\SN
SNLS
Union
Constitution
SN2009
m
+0.10
-0.08
+0.05
-0.19
-0.13
+0.07
-0.10
w0
+0.06
+0.13
-0.04
+0.09
+0.08
-0.05
wa
+0.24
-0.50
+0.53
-0.24
+0.40
-0.58
+0.34
-0.35
Statistic
(0.07)
(0.7)
(10.0)
Bias well below statistic errors for wa and wa
Higher bias for m. But this is an artifact of Fisher analysis error estimate!

22. (WMAP+SNIa+WL)%BAO
Try to predict the space 2-points correlation function (r) of Luminous red Galaxies (SDSS3). The BAO bias is fixed from the 4-probes combination
Good agreement below 70 Mpc
Too much BAO signal at 150 Mpc
1 sigma effect
Better agreement with a lower mh2 but then inconsistency at 40 Mpc scale! (Eisenstein 2005)
If effect due to systematic, significance of the BAO peak is lowered down to 2.4 sigma.

23. (WMAP+SNIa+BAO)%WL
Use the WMAP+SN1a+BAO to predict the aperture mass square of weak lensing
Good agreement for 8<<70 arcmin. B mode compatible with 0
Signal larger at small scale <2 arcmin
-Can be due to non linear effects to the matter power spectrum (~5-10%)
-Systematic as indicated by B-mode
Bump at 120 arcmin (2o).
At most a 1.6  effect.
B mode is significantly not 0
Discrepancies observed follow what B mode tell us. But not very significant.

24. (SNIa+BAO+WL)%WMAP
We try to predicted Cl’s with the 3 other probes.  is fixed by the 4 probes best fit.
-As expected, error from other probes is huge compare to CMB
-However, the Cl’s are out of prediction ~1 sigma effect
-Mainly due to the high value of m from SN+best fit m=0.450.08
BAO(CDM) : b =0.0460.01
BAO+SN: b =0.0630.024
Correlation b% m=93%
Hao Liu & Ti-Pei Li: b =0.0520.03

25. Summary
We have developed a new statistical approach to combine probes.
Results are compatible with previous published results.
We confirm we are living in CDM Univers
We are at a level where we can start systematic study using 3 probes to predict a 4th one
All probes follow what is predicted by others (1)
No significant evolution of SN magnitude with redshift observed (8%) systematic lower than statistic error
Clearly, next data from Planck and from other surveys will help us to constraint systematic and/or new physic (WL or BAO tomography)

26. What about other surveys
There is still systematic in weak lensing

27. SN2009+BAO+WL SN2009 alone : m = 0.18 (35) wa = 1.  (80)
SN2009+BAO+WL: b= 0.0650.02 wa=0.470.7
Full SN contour
SN alone
Prefer SN value for wa=0.47

28. Probability density of the 2
Assume we repeat N times the measurement of x and x has a Gaussian pdf g(x).
For each measurement we can define a 2 :
The confidence level is defined as: The confidence interval as:

29. MCMC in few words
MCMC explores the Likelihood with respect to cosmological parameters using a chain of correlated points.
It’s a random walk using metropolis method.
How to simulate a set of points such that
the points density is proportional to a given pdf ?
Start from x0 xi
Simulate a random step s and compute xi+1=xis
If p(xi+1)>p(xi) keep the point xi+1 xi goto 2
If not we simulate a random number (r) between 0 and p(xi)
If r>p(xi+1) we reject the point and goto 2
If r<p(xi+1) we keep the point xi+1 xi goto 2
The central limit theorem tell us that for N the set of points {xi} converge toward the true pdf. (typically 1 million for MCMC in cosmology)

30. What’s the difference with the Montecarlo Frequentist Method ?
The frequentist MC randomizes observables (Cl’s, magnitude…) is such a way that each randomization has the correct probability (no chain)
Original version of metropolis method:
1) Select a random point x
2) Select a random point y between [0.,pmax]
3) If r>p(x) reject the point and go to 1
4) If r<p(x) you get it. End
A point x is selected according to its probability.
No problem of convergence!
The frequentist MC simulates experiment independently (uncorrelated.)
In case of correlation between observables, use the experiment covariance matrix to go to the diagonal space, do the simulation and then go back to physical observables…

31. What about marginalization?
In MCMC approach, the marginalization is obtain by projection of the likelihood:
{mi,I,w0i,wai}likelihood {w0i,wai}likelihhod equivalent to an integration of the likelihood over {m,}  average
In frequentist MC, marginalization is a projection of the most likely simulated experiments.(2  2min + 2)
{mi,I,w0i,wai}best_fit {w0i,wai}best_fit  most probable
If errors on ’s Gaussian ->equivalent method: mean = max likelihood.
So, what’s the best method?
It’s mainly a philosophical question. Keep both methods and compare.
What means the best?
The most probable. Fine, but it’s sensitive to the statistical realization of the experiment and secondary maximum of Likelihood.
The average. It’s closer to what we expect from the experiment (no statistical realization). Fine, but in that case, we don’t use all statistical power of the experiment.

32. Marginalization example
Frequentist
MCMC

33. Union sample
BAO+UNION+WL: wa = 0.81