Aucun titre de diapositive

1
Multi-probe determination of cosmological
parameters Prospects for Dark Energy Evolution
Christophe Yèche (CEA-Saclay)
on behalf of A.Ealet, A. Réfrégier, C. Tao, A.
Tilquin, J.-M. Virey, and D. Yvon.
Sino-French Workshop on the Dark Universe,
Marseille, September 19-23, 2005
2
Outline

• Statistical approach
• Frequentist Vs Bayesian.
• Priors in Bayesian approach.
• Our method a frequentist approach.
• A method for prospective study Fisher matrix
  technique.
• Combination of current surveys with a
  frequentist approach
• Data considered WMAP SNIa (Riess 2004).
• Impact of DE perturbations on results for w.
• Validation of Fisher matrix technique.
• Combination of the future surveys
• Sensitivity studies performed for three
  scenarios
• Today, Mid Term (2006-2007) and Long Term
  (2013-2015).
• Combination of three different probes CMB, SNIa
  and WL.

2
3
Statistical Approach
3
4
An example of what we want to do

• Use the published data (for instance, WMAP)
• Test the method in fitting WMAP data with a flat
  adiabatic ?-CDM model with 6 parameters (?bh2,
  ?mh2, h, A(?82), ns, ?)
• Compare to published results by WMAP
• Final Goals
• Develop a flexible tool to extract Confidence
  Level (CL) on the cosmological data based on a
  frequentist approach.
• Possibility to add new probes SNIa, WL
• Possibility to add new variables (w)

Parameters measured by WMAP ?bh2 0.024 ?
0.001 ?mh2 0.14 ? 0.02 h 0.72 ? 0.05 A 0.9
? 0.1 ns 0.99 ? 0.04 ? 0.166 ? 0.08
4
5
Frequentist approach Vs Bayesian approach

• Frequentist approach
• The frequentist approach returns the region
• of the parameters for which the data have at
  least
• a given probability (1-CL) of being described
• by the model.
• We assume a true value ? (optical depth)
• and we measure the agreement of the model
• for this given value.
• Bayesian approach
• The bayesian approach tries to calculate
• the probability distribution of the true
  parameters
• by assigning probability density functions
• to all unknown parameters.
• The method wants to predict a value
• (much more ambitious than frequentist)
• Based on Bayes theorem!

1-CL
WMAP Team Official paper
5
6
Prior in bayesian approach

• Marginalization of the Likelihood
• To get a PDF on the cosmological variable
• the marginalization of the likelihood
• (integration over other variables) is performed
• with Bayes Theorem.
• Priors are needed in Bayesian approach.
• Priors are completely arbitrary.
• A prior for ?
• Assuming a flat prior on ? between
• 0 and 0.3 seems reasonable (for instance,
• done by WMAP team in their paper).
• Actually, it may have a very large
• impact on other cosmological parameters
• Suppression of acoustic peak,
• z at reionization or time of reionization

6
7
Example of the impact of a prior choice

• Simplified example
• Two parameters A and ?.
• A (?82) is measured with TT and TE spectra.
• ? depends only on the first point
• of the TE spectrum.
• Likelihood marginalized for different
• priors for ?.
• PDF(A) depends a lot on ? prior choice !

7
8
A Fitter with Frequentist approach

• Principle
• Minimize a ?2 defined as
• Determine the minimum ?02
• Fix one of the variables, for instance ?I
• Repeat the fit and determine the new minimum
• Confidence Level f parameter ?i is defined by
• A scan of ?i gives the C.L. distribution
• Extension to 2-D contours with two fixed
  parameters ?i and ?j.
• Technical implementation
• Use Minuit as Fitter, cmbeasy for computation of
  Cl and Kosmoshow fot SNIa ?2 .
• Minimization obtained after 200-300 call to
  cmbeasy.
• One job per batch (1h) at CCPN (IN2P2 computing
  Center)

1-CL
CL
8
9
Results and Examples of 1D Confidence Level Plot
WMAP ?bh2 Scan
?bh2 0.0215, 0.0239_at_68 0.0209,
0.0247_at_90 ?mh2 0.130, 0.162_at_68
0.121, 0.174_at_90 h 0.66, 0.75_at_68
0.63, 0.78_at_90 A 0.75, 0.93_at_68 0.70,
1.00_at_90 ns 0.958, 1.013_at_68 0.943,
1.034_at_90 ? 0.091, 0.0173_at_68 0.068,
0.204_at_90
WMAP h Scan
9
10
A method for prospect studies Fisher matrix

• In principle with the covariance matrix we can
  have an estimate of the statistical errors.
• Pros Extremely fast method, Just need to
  compute derivative for each variable
• Cons Raw approximation when the variable are
  very correlated or the ?2 is non-parabolic.

Derivatives
10
11
Examples of 2-D Confidence Level Plot
WMAP ?bh2/ ?mh2 Scan with frequentist approach
WMAP ?bh2/ ?mh2 Scan with Fisher matrix

• More realistic CL contour with frequentist
  approach
• In general, Fisher matrix technique is
  satisfactory in many cases
• except (w0, wa)..see next slides

11
12
Combination of current surveyswith a
frequentist approach
12
13
Data Set
Riess et al., 2004, Astrophys. J. 607, 665
Hinshaw et al., 2003, Astrophys. J. Supp. 148, 135

• WMAP TT and TE spectrum, without correlation
  matrix between the Cl (We checked with the
  official WMAP likelihood that the results are not
  affected by this assumption).
• 157 SNIa Riess et al. 2004 and a few from the
  HST
• combined with the kosmoshow program.

13
14
Cosmological parameters

• Concordance Model (?-CDM Model)
• ?b/?m density for baryon/matter, h Hubble
  constant,
• ns spectral index, ? reionisation optical depth
• A(?82) normalization parameter for CMB (and
  WL).
• Ms0 normalization parameter for SNIa.
• Parameterization of Dark Energy
• z-dependent wp/? to take into account dark
  energy evolution.
• w(z) parameterization has to fit all the data
  over a large range from z0-2 (SNIa) to z1100
  for CMB.
• ww0 for t0 and ww0wa for t?0
• Unphysical region for w0wagt0
• Dark Energy perturbations for CMB
• Tricky part for CMB, we have considered 2
  options
• No perturbation DE homogeneous and no
  clustering
• With perturbations only for wgt-1 (otherwise,
  even t GR is affected!!!).

14
15
Fit Results for w0 and wa

• Value of w0 smaller than -1.0 are favored (in
  particular we add DE perturbations).
• Discrepancy in w0 according the strategy for DE
  perturbation treatment.
• Errors significantly degraded when DE
  perturbations added.

15
16
Comparison with previous combinations

• Two sets of results
• 1) Seljak et al. astro-ph/0407372
• with w0 -1 and wa0.
• 2) Upadhye et al. astro-ph/0411803
• with w0 -1.3 and wa1.2
• Our study indicates that this
• discrepancy may be due to a
• different treatment of DE
• perturbation in CMB.
• Detailed reading of these papers
• seems to confirm this hypothesis.
• No DE perturbations
• DE perturbations for wgt-1.

Upadhye et al., astro-ph/0411803
16
17
Comparison of the Errorswith Fisher Matrix
technique

• In the rest of the study we switch off
• the DE perturbation in cmbeasy code.
• The comparison of the errors at
• 68CL for the frequentist approach
• and the Fisher matrix technique are
• very good even for w0 and wa (not
• true for 95CL).
• Demonstration that the Fisher matrix
• technique will be a reliable method for
• the prospect study.

17
18
Combination of future surveyswith Fisher Matrix
18
19
Three Scenarios

• Today
• Reference point
• WMAP 157 SNIa (Riess 2004).
• Mid-term
• 2007-2008 (before Planck)
• CMB WMAP Olimpo (balloon experiment with good
  resolution and a small field).
• SNIa SNLSSNFactoryHST
• WL CFHT-LS.
• Long-term
• 20112-2015
• CMB Planck
• SNIa SNAP
• WL SNAP

19
20
Error on Cl with Olimpo
Cosmic variance
Detector performances
Cosmic variance
Resolution effect

• Nominal configurations
• Observed Sky S300 deg2
• Detector sensitivity s150 ?Ks1/2
• Number of bolometers Nbolo 20
• Observation time Tobs 10 days
• Resolution ?fwhm 4 arcmin

20
21
Error on Cl with Planck

• Nominal configurations
• Full Sky, 12 months
• 3 Frequencies 100 / 143 / 217 (GHz)
• Sensitivity R 2.0 / 2.2 / 4.8 (?K/K)
• Resolution ?fwhm 9.2 / 7.1 / 5.0 arcmin
• (Figures extracted from The Planck mission J.A.
  Tauber
• JASR 6394 (2004) )

21
22
Stronger constraints thanks to correlations

• For ?m-w0 and for w0-wa, we observe
• orthogonal correlations between (CMB-WL)
• and SNIa
• It allows to break the degeneracies.

• Actually we have 9-dim correlations,
• we gain more than the simple 2-dim
• overlap!
• We use the full correlation matrix.

22
23
Expected sensitivity

• With ground observations (mid-term), good
  precision on w0 alone, but not on wa.
• Satellite observations (Planck and SNAP) are
  required to achieve the 0.1 precision on w.
• Very impressive sensitivity on other parameters,
  in particular ns (test of inflationary models).

23
24
Expected sensitivity for Long Term Scenario
SUGRA
?-CDM
Phantom

• With precision expected for Long-term scenario,
  we can disentangle 3 extreme models in w0-wa
  plane (Phantom - ?CDM SUGRA).
• Sensitivity depends on the position in the w0-wa
  plane.

• In the Long-term scenario, WL gives very
  promising results (as good as SNCMB).
• However, systematics effects not studied for WL.

24
25
Conclusion andProspects
25
26
Conclusions

• We have developed a frequentist approach to
  treat the current data (WMAP SNIa), we get for
  the DE evolution
• We have observed a strong effect on (w0, wa)
  related to the treatment of DE perturbations,
  which may explain the discrepancies observed in
  the literature.
• Our prospect study demonstrates that we need
  satellite observations (long term scenarios) to
  achieve a 0.1 sensitivity on wa.
• Finally, we can easily add new probes in our
  tool.
• Further information in the paper submitted to
  AA
• Ch. Yèche, A.Ealet, A. Réfrégier, C. Tao, A.
  Tilquin, J.-M. Virey, and D. Yvon,
  astro-ph/0507170.

26
27
BackUp Slides
27
28
Cross-checks of basic hypotheses
?2 for WMAP

• Simulation of 300 experiments similar to WMAP.
• Distribution of ??2 for fits with 6 free
  parameters
• and 4 free parameters.
• Distribution of ??2 for fits with 6 free
  parameter fits
• and initial ?2.
• For both the ??2 distributions fit well with
  the ?2
• distribution with 2/6 degrees of freedom.

• Simulation of 300 experiments
• similar to WMAP.
• The cosmological model works
• Rather well.
• Probability of 25 to get a greater
• value for ?2

28
29
Limit on neutrinos mass and test inflation
models
Confidence level for (f?, ?mh2) marginalized on
(?bh2, ?mh2, h, A, nS, ?)
Confidence level for (r, nS) marginalized on
(?bh2, ?mh2, h, A, ?)
29