Sensitivity and figures of merit for Dark Energy Supernovae surveys Jean-Marc Virey Centre de Physique Théorique & Université de Provence, Marseille 2 nd 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/

Introduction The Universe is accelerating : * Many Observations : SNIa, CMB, LSS (P(k), BAO, WL, …) * Many Theoretical Interpretations : m Quintessence : φ ->V(φ), φ in, φ in ì Cosmological constant / Vacuum energy : X=Λ => w x =w Λ =-1 ρ Λ (z)=ρ Λ 0 ì Modifications Friedmann eq. / General Relativity : ì Scalar fields : m More exotic : k-essence, phantom, quintom

ì Perfect fluid : ρ X (z), w X (z)=P X /ρ X Too many possibilities => * Try a model independent approach : * Assume DE= Λ ì Use a parameterization : eg Linder/Polarski : w(z)=w 0 +w a (1-a) =w 0 +w a z/(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), ρ X (z)/ρ X (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)=w 0 +w a z/(1+z) combined analysis from Zhao,Xia,Feng,Zhang (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= z max =1 : Stage 2 : ground, near future (SNLS type … 2010) b) N= z max =1 : Stage 3 : ground/space, future (DUNE type … 2015) c) N= z max =1.7 : Stage 4 : space (infrared), future (SNAP … >2015) d) N= z max =1.7 : Stage 5 : space (infrared), far future (JEDI t. … ??)

Hypothesis for the analysis :  Fiducial model : ΛCDM  flat universe : Ω T =1 exactly  strong Ω M prior : Ω M =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 : 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 case

The DETF figure of merit The pivot parameterization : with and Properties : * both pivot and Linder parameterizations are equivalent * w 0 and w a are decorrelated at z p * «sweet spot» at z p ie σ(w p ) is the smallest * σ(w p ) = σ(w=constant) * contours in (w 0,w a ) and (w p,w a ) planes are mathematically equivalent * zp value depend on all the details of the analysis => NO physical significance... DETF figure of merit   inverse area of (w 0,w a ) error ellipse

Is z p meaning something ? z p value very dependent on : * data model (ie N and z max ) * presence of systematic errors and also (not shown) : * Nearby sample * Ω M prior * fiducial model conclusions : * z p is not representative of any physical characteristics of the survey nor of the DE dynamics ! * comparing w p constraints from  data model is ambigous => (w p,Ω M ) and (w p,w a ) contours should be interpreted with cautions * However, no problem for the DETF values (  inverse area of (w 0,w a ) 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 Results : * stat : prefer high N than deep z max stage3 better than stage4 * syst : prefer deep z max than high N stage’s chronology respected highest 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 !!!

z max optimisation Interplay between N and z max very dependent on the systematics => for fixed N and systematics we study the optimisation of z max see also Linder&Huterer 03, PRD DETF figure of merit ratio for adjacent z-bin (N=cste) ie effect of adding 1 z-bin of width 0.1 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. : z max  1.7  N stat. : z max   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 & wp black : stat & wa ; red : stat & wp plain : N=2000 ; dotted : N=15000 Results :  syst. : no evolution with N if gain > 5% then : wa : zmax  1.7 wp : zmax  0.9 (higher N increase very slightly z max )  stat. : strong evolution with N higher N decrease strongly z max 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 but obscure for N evolution => look at the DETF figure of merit directly (ie not a ratio) syst. stat. N=15000 N=2000 N=15000 Results :  DETF figure of merit always increases with N or zmax : syst. : small improvement with N stat. : 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 δm syst dependent  If : a higher z max is preferable (ie z max >1.7), and the N dependence is reduced  If systematic errors are negligeable, N is the fundamental parameter and z max around 1 is sufficient  With correlated systematic errors (ie z-dependent ), one gets similar results

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 models more 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, (2005) Barger et al., Phys. Lett. B635, 61 (2006)

δm syst =0.02 wa w0 black : stage 2 green : stage 3 red : stage 4 blue : 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: illustration excluded 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 z p (w p 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 :

Informations :  get the z-dependence of the constraints Re: 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 z max 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 N  Gain comes from the increase of the survey depth up to z max =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

 (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)