Sensitivity and

figures of merit for

Dark Energy Supernovae surveysJean-Marc Virey Centre de Physique Théorique & Université de Provence, Marseille

2nd Sino-French workshop on the « Dark Universe »

September 2006 Beijing

Summary

Introduction and SNIa data models

Sensitivity of the DETF figure of merit

(optimisation/comparison of SN surveys)

Figures of merit and DE models distinction

Based on J.-M. Virey and A. Ealet, astro-ph/0607589

Introduction

The Universe is accelerating :

* Many Observations : SNIa, CMB, LSS (P(k), BAO, WL, …)

* Many Theoretical Interpretations :

Quintessence : •φ ->V(φ), φin, φin )(

)(2

2/1

2

2/1

V

Vweff

Cosmological constant / Vacuum energy : X=Λ => wx=wΛ=-1 ρΛ(z)=ρΛ0

Modifications Friedmann eq. / General Relativity :

Scalar fields :

More exotic : k-essence, phantom, quintom

320

22

)/(ln

ln311 aHHH

adHdw M

eff

Perfect fluid : ρX(z), wX(z)=PX/ρX

Too many possibilities =>

* Try a model independent approach :

* Assume DE= Λ

Use a parameterization : eg Linder/Polarski : w(z)=w0+wa (1-a) =w0+waz/(1+z)

+++ : model «independent» , simplicity, fast calculations----- : intrinsic limitations, analysis bias, «theoretical bias»

Re: other «model independent» approaches (with own pb’s): other parameterizations, other basic observables (H(z), q(z) (kinematic), ρX(z)/ρX(0) (dynamic)) or non-parametric tests

Whatever the method is : * the Universe is accelerating * Λ is OK at 68%CL (at the boundary)

Linder/Polarski : w(z)=w0+waz/(1+z)combined analysis from Zhao,Xia,Feng,Zhang 0603621 (SNIa+CMB+LSS(P(k),Ly-α))

SN Data Models

Experimental side : many new SN surveys are proposed

figures of merit are needed to compare the sensitivities (errors on w(z) param.)

Data models :

a) N=2000 + zmax=1 : Stage 2 : ground, near future (SNLS type … 2010)

b) N=15000 + zmax=1 : Stage 3 : ground/space, future (DUNE type … 2015)

c) N=2000 + zmax=1.7 : Stage 4 : space (infrared), future (SNAP … >2015)

d) N=15000 + zmax=1.7 : Stage 5 : space (infrared), far future (JEDI t. … ??)

Hypothesis for the analysis :

Fiducial model : ΛCDM flat universe : ΩT=1 exactly strong ΩM prior : ΩM=0.27±0.01 (WMAP-3 central value, Planck error)

«Nearby sample» (SN Factory) : 300 SNIa at z<0.1

magnitude dispersion of 0.15 :

systematic errors :

binstat Nm /15.0

02.0systm222

syststat mmm

Re: many systematic cases studied with z-dependence and amplitudes

Results are very dependent on the amplitude is already optimistic

in the following : comparison of statistical case with case02.0systm

02.0systm

The DETF figure of meritThe pivot parameterization :

with and

Properties :

* both pivot and Linder parameterizations are equivalent

* w0 and wa are decorrelated at zp

* «sweet spot» at zp ie σ(wp) is the smallest

* σ(wp) = σ(w=constant)

* contours in (w0,wa) and (wp,wa) planes are mathematically equivalent

* zp value depend on all the details of the analysis => NO physical significance ...

00

00

wwawwa

wwawp C

Cz

zw

zwwwaawzw a

p

apapp

11

)()(

p

papp zz

wwzww

1

)( 0

DETF figure of merit inverse area of (w0,wa) error ellipse 1)()( ap ww

http://www.nsf.org/mps/ast/detf.jsp

Is zp meaning something ?

zp value very dependent on :* data model (ie N and zmax)* presence of systematic errors

and also (not shown) :* Nearby sample* ΩM prior* fiducial model

conclusions : * zp is not representative of any physical characteristics of the survey nor of the DE dynamics !* comparing wp constraints from data model is ambigous=> (wp,ΩM) and (wp,wa) contours should be interpreted with cautions* However, no problem for the DETF values ( inverse area of (w0,wa) error ellipse)

DETF figure of merit for the SN data models

We plot «normalized» figure of merit : it allows a direct interpretation of the scale

stageXapstage

ap wwww )()(/)()( 2

Results :

* stat : prefer high N than deep zmax

stage3 better than stage4

* syst : prefer deep zmax than high Nstage’s chronology respectedhighest slope between stages 3 and 4

conclusions : the level of systematic errors is the key discriminant between wide or deep surveys reinforces the need for a deep survey stages 3 and 4 equivalent if !!!

02.0systm

006.0systm

zmax optimisation

Interplay between N and zmax very dependent on the systematics =>for fixed N and systematics we study the optimisation of zmaxsee also Linder&Huterer 03, PRD67 081303

DETF figure of merit ratio for adjacent z-bin (N=cste)

ie effect of adding 1 z-bin of width 0.1

max1.0max )()(/)()( zap

zap wwww

syst.

stat.

N=2000

N=15000

Results : low z : net improvement high z: saturation, dep. on N+syst.

5% gain if gain > 5% then : syst. : zmax 1.7 N stat. : zmax 1.1 - 1.2 N

Re : the weak dependence on N for the stat case is accidental :wa and wp exhibits strong variations with N for the zmax evolution,which cancel eachother when looking at the DETF figure of merit

blue : syst & wa ; green : syst & wpblack : stat & wa ; red : stat & wpplain : N=2000 ; dotted : N=15000

Results : syst. : no evolution with Nif gain > 5% then : wa : zmax 1.7 wp : zmax 0.9(higher N increase very slightly zmax)

stat. : strong evolution with Nhigher N decrease strongly zmax if gain > 5% then : wa : zmax 1.3 or 1 wp : zmax 0.6 or <0.4 !!! (special)

Impact of N

Previous figure with DETF ratio very helpful for zmax evolution butobscure for N evolution => look at the DETF figure of merit directly(ie not a ratio)

syst.

stat.

N=15000

N=2000

N=2000

N=15000

Results : DETF figure of merit always increases with N or zmax :syst. : small improvement with Nstat. : strong improvement with N

the impact of N is directly connected to the level of systematic errors

Conclusions of part 1 (SN surveys comparison) :

If : * SN at z>1 are mandatory to increase the constraints * With SN at z>1.7, the improvement is marginal * these results are N independent but δmsyst dependent

If : a higher zmax is preferable (ie zmax>1.7), and the N dependence is reduced

If systematic errors are negligeable, N is the fundamental parameter and zmax around 1 is sufficient

With correlated systematic errors (ie z-dependent ), one gets similar results

02.0systm

02.0systm

Figures of merit and DE models distinction

Previous question : sensitivity of the SN surveys (from the DETF f.o.m.)

Present question : capability of the SN surveys to separate DE modelsmore precisely : which DE models are in agreement with ΛCDM ?

To compare DE models we need not only the error on the parameters but also their central values.

We concentrate on 2 figures of merit :

* contours in (w0,wa) plane

* representation of w(z) with error shape variations with z

Contours in the (w0,wa) plane : helpful informations ?

+++ : * recover previous results on data models comparison * possibility to put classes of DE models ---- : the plane is defined from an adhoc parameterization

Caldwell-Linder PRL95, 141301(2005) Barger et al., Phys. Lett. B635, 61 (2006)

δmsyst=0.02

wa

w0

black : stage 2green : stage 3red : stage 4blue : stage 5

Separating DE models :

Q: - we want to exclude a particular DE models from ΛCDM - which data model optimisation is best for that ?

Answer : It depends strongly on the DE model ...

models A-D: illustrationexcluded at 95%CL with stage 4

* A exclusion indep. of the data model specifications => no sensitivity along short axis

* better the survey, better exclusion for B,C,D => sensitivity along large axis

SN survey optimisation reduces degeneracies among w0 and wa which is represented by a reduction of the large axis and it has no impact on the short axis

* A is excluded thanks to the constraints at zp (wp best «observable»)

* B : low z + high z behaviours (w0 + wa)

* C : low z (w0)

* D : high z (wa)

2σ errors for ΛCDM

w(z) with error shape variations with z : helpful informations ?

Errors with the w0-wa parameterization :)1(

)()(2)1(

)()())(( 002

22

022

zzwwC

zzwwzw awawa

Informations :

get the z-dependence of the constraintsRe: different from the z location of the SN providing the constraints

see the sweet-spot at zp explicitely

wp only cannot be used to separate Λ from all DE models

Cautions :

error shapes are parameterization dependent (ie potential bias)

strong correlations among parameters and among z-bins => existence of artefacts if not used in the realistic z-range probed by data

Interesting property :

various strategy of analysis (eg non parametric tests, other parameterizations …) may be confronted in this plane =>

useful for consistency checks …(?)

We can get subtle details by interpreting both representations :

models along the degeneracy line maybe distinguished from ΛCDM by the low and high redshift behaviour of the equation of state as encoded in the w0 and wa parameters, whose expected precision depend strongly on the SN survey configuration

models «orthogonal» to this line may be excluded thanks to the constraints at the pivot redshift, whose expected precision depend weakly on the SN survey configuration but more on the control of the systematic errors

Conclusions

DETF figure of merit : good approach to test SN surveys optimisation

N and zmax optimisations are very dependent on the level of systematic errors. Conclusions depend on the control of sytematic errors which fix the amount of information inside data. Ignoring them can provide some biases and increasing the statistics will only increase the statistical error on a wrong cosmology.

For , no extra information for N2000. Gain comes from the increase of the survey depth up to zmax=1.7. This conclusion is quite strong and only change for a statistical error only scenario.

DETF drawback : lack information to estimate discrimination power between DE models

02.0systm

(w0,wa) contour plot : better understanding and good discrimination, since classes of models can be put in this plane

(w,z) plane : complementary information : redshift dependence Re : the results are parameterization dependent, but this plane may be used for consistency cheks

Data models comparison : some degeneracies remain even for the most ambitious projects (ie some classes of models (eg freezing models) cannot be better constrained by increasing SN surveys performances)

solutions ? Yes : combined analysis. But : be careful since systematic errors will dominate future analysis and will introduce stronger bias in a combination =>* check compatibility between probes in a coherent way (same theoretical assumptions, framework, teatment of systematics)* combinations 2 by 2 should help the control of the systematic effects (and also to cross check internal hypothesis and to control the results)