d nfw s irène balmÈs

TIME-DELAYS ANDSPARSITY

IRÈNE BALMÈS

TIME-DELAYS

TIME-DELAYS

BAYESIAN ANALYSIS

ANGULAR STRUCTURE

TEST

RESULTS

SPARSITY

DARK MATTER HALOS

DEUS

NFW PROFILE

SPARSITY

ALTERNATIVE PROBES OF COSMOLOGY:TIME-DELAYS IN STRONG GRAVITATIONAL

LENSES AND SPARSITY OF DARK MATTER

HALOS

Irène BALMÈS

March 27th 2014 – extragalactic seminar

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A PICTURE OF THE UNIVERSE

STANDARD MODEL OFCOSMOLOGY

I baryons: ∼ 5%

I dark matter: ∼ 25%

I dark energy: ∼ 70%

Cosmological MicrowaveBackground

Large ScaleStructures

Supernovae

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QUESTIONS IN COSMOLOGY

What is dark matter?What is dark energy? Is it a simple cosmological constantor a quintessence field?Is general relativity the correct description of gravitation?Is the Universe homogenous and isotropic?

COSMOLOGICAL PROBES

Cosmological Microwave Background (COBE, WMAP,Planck), luminosity distances (UNION dataset), BaryonicAcoustic Oscillations (SDSS, 2dF, 6DF, Wiggle-z, BOSS),number of structures (XMM-LSS)

put constraints on H0, Ωm, ΩDE , w ...

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DARK MATTER HALOS

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DARK MATTER HALOS

Result of the growthof initialperturbationsinfluenced bygravity andexpansion.

Traced byobservablegalaxies, lensingeffect,...

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1 TIME-DELAYS AS A COSMOLOGICAL PROBETime-delaysBayesian analysisIndependence from the angular structureTesting on synthetic catalogsResults from observational data

2 SPARSITY OF DARK MATTER HALOSDark matter halosDEUSAgreement to the NFW profileSparsity: an alternative parameter

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WHAT IS A GRAVITATIONAL LENS?

DENOMINATION

2 images : double lens4 images : quadruple lens

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IRÈNE BALMÈS

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S

Obs

I

Lθ

β

α

I′

FERMAT’S PRINCIPLE

Images are formed atextremum of the arrivaltime.

β: source positionθ: image position onthe skyα: light deviation

∆t = (1 + zl)dlds

dls

[12

(θ − β)2 − ψ(θ)

]

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TIME DELAY SURFACE

source plane

lens plane

observer view

time arrival surface

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IRÈNE BALMÈS

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TIME DELAY SURFACE

source plane

lens plane

observer view

time arrival surface∆t

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IRÈNE BALMÈS

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TIME DELAY EXPRESSION

∆tA,B = (1 + zl)dlds

dls

12

[(θA − β)2 − (θB − β)2

]+ ψ(θA)− ψ(θB)

with

dlds

dls∝ H−1

0

I ∆tAB: measured when the source is variable (quasar)I mass distribution: reconstructed through a model

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SPHERICAL MODEL

ρ(r) ∝ r−n

ψ(~θ) =θ2

E3− n

(θ

θE

)3−n

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ELLIPTICAL MODEL

ψ(~θ) =θ2

E3− n

√

(1− ε)θ21 + (1 + ε)θ2

2

θE

3−n

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INFLUENCE OF THE ENVIRONMENT

SEVERAL COMPONENTS

ψ(~θ) = ψ0(~θ) + ψ1(~θ) + ....

or

GLOBAL SHEAR TERM

ψ(~θ) = ψ0(~θ)− 12γθ2 cos 2(φ− φγ)

High number of parameters (θE , n, ε, φε, γ, φγ ...): modelsunderconstrained.

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MODEL DISCRIMINATION

χ2 COMPARISON

χ2 =1n

n∑i=1

(d iobs − d i

pred(p))2

2σ2i

only compares one point ofthe parameter space

EVIDENCE COMPARISON

E =

∫1√2σ2

e−χ2(p)dp

compares the overallprobability on the wholeparameter space

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IMPORTANCE OF THE ANGULAR STRUCTURE

If the images lie on a diameter (|δ| << π), the time-delayis insensitive to the angular structure of the lens potential.Meylan et al, 2006

lens

A

B

δ

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IMPORTANCE OF THE ANGULAR STRUCTURE

Asymmetrical images (|θA − θB| > 0.2(θA + θB)) createtime-delays insensitive to shear and substructures.Oguri, 2007

lensθA

A

B

θB

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TWO COMPETING MODELS

SPHERICAL MODELM0

ψ(~θ) =θ2

E3− n

(θ

θE

)3−n

MODEL INCLUDING ANGULAR STRUCTUREM1

ψ(~θ) =θ2

E3− n

(θ

θE

)3−n

− 12γθ2 cos 2(φ− φγ)

Use of Bayesian evidence to determine whether the datarequires modelM1.

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TESTING OUR METHOD

values of ∆tBayesianmodel selection

subsampleof lenses S0

Bayesianparameter estimationrecovered H0

assumed H0

synthetic lens catalog

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CATALOG WITH SHEAR

500 lenses simulated using gravlens (Keeton, 2011)with modelM1:

ψ(~θ) =θ2

E3− n

(θ

θE

)3−n

− 12γθ2 cos 2(φ− φγ)

RANDOMLY GENERATED PARAMETERS

I θE = 1I n = 2± 0.3I γ = 0.2± 0.1I φγ = 0I source position: −1 < x , y < 1

→ 280 double lenses

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BAYES FACTOR

favorsM0

spherical model

favorsM1

model with shear

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RECOVERED VALUE OF H0Starting value for h = H0/100 km/s/Mpc: 0.72

RESULT FROM THE ANALYSIS OF S0

h = 0.76+0.016−0.015: compatible within 3σ with the starting

value.19 / 46


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ANALYSIS OF S1 AND S0 + S1 WITHM0

Results heavily biased towards high values of h.

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ANALYSIS OF S0 + S1 WITHM1

Degeneracies lead to highly biaised value of h.

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CATALOG WITH ELLIPTICITY

500 lenses simulated using gravlens and with ellipticityinstead of shear

ψ(~θ) =θ2

E3− n

√

(1 + ε)θ21 + (1− ε)θ2

2

θE

3−n

RANDOMLY GENERATED PARAMETERS

I θE = 1I n = 2± 0.3I e = 0± 0.2I φe = 0I source position: −1 < x , y < 1

→ 307 double lenses

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BAYES FACTOR

S0: sample of lenses described byM0S1: sample of lenses described byM1

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RECOVERED VALUE OF H0

Starting value for h: 0.72

S0: h = 0.771± 0.014

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OBSERVATIONAL DATA

LENSES WITHMEASURED ∆t

I 12 doublesI 8 quads

BAYES FACTOR SELECTION

3 lenses favor modelM0:I B1600+434I SBS 1520+530I SDSS J1650+4251

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BAYESIAN PARAMETER ESTIMATION

For each lens i : determination of ni , Ωm and H0

L(ni ,Ωm,H0|α) =

∫∫f (~ΘA, ~ΘB)L(ni ,Ωm,H0|~ΘA, ~ΘB, α) d~ΘAd~ΘB

f (~ΘA, ~ΘB) = exp

−∑A,B

(θi − ~Θi)2

2σ2i

Combination of all lenses: marginalizing over ni

L(Ωm,H0) =∏

i

∫L(ni ,Ωm,H0)dni

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RESULTS ON H0 AND Ωm

Ωm is not constrained efficiently by our sample. Morelenses are needed.

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RESULTS ON H0

H0 = 76+15−5 km/s/Mpc

results published in Balmès & Corasaniti, 2013

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1 TIME-DELAYS AS A COSMOLOGICAL PROBETime-delaysBayesian analysisIndependence from the angular structureTesting on synthetic catalogsResults from observational data

2 SPARSITY OF DARK MATTER HALOSDark matter halosDEUSAgreement to the NFW profileSparsity: an alternative parameter

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A UNIVERSAL PROFILE?

NAVARRO, FRENK ANDWHITE PROFILE

Phenomenological densityprofile from the simulations

ρ(r) =ρs

rrs

(1 + rrs

)2

concentration: c = r200rs

Navarro, Frenk & White, 1997

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DEUS SIMULATIONS

I multiple box-sizesand resolutions

I large mass rangeI several

cosmologicalmodels

see www.deus-consortium.org

Ideal set-up to studyimprints of cosmologyon non-linear structureformation, and inparticular on haloprofiles.

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www.deus-consortium.org

www.deus-consortium.org


IRÈNE BALMÈS

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COSMOLOGICAL MODELS

REALISTIC MODELS

calibrated on WMAP5 (3,1) andthe UNION dataset

I ΛCDMW5 (3, 1)I SUCDM:

w ' −0.94 + 0.19(1− a)

I RPCDM:w ' −0.87 + 0.08(1− a)

TOY MODELS

Study of theinfluence of onesingle parameter

I SCDM:ΩΛ = 0

I LΛCDM:ΩΛ = 0.9

I LRPCDM:equation ofstate w

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HALO FINDERS

OBSERVATIONALLYDRIVEN

Spherical OverdensityHalo: sphere of meandensity ∆× ρm∆ = 200imposed sphericalgeometry

PHYSICALLY DRIVEN

Friend of FriendHalo: group of linkedparticles.no imposed geometrymerging halos risk beinglinked

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PROFILE FITTING

>From SOD: M200, r200, ri , ρi , σi

Fitting procedure: c, χ2 = (n∑

i=1(ρNFW

i − ρi)2/2σ2

i )/n

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NUMERICAL EFFECT ON χ2

Renormalization: χ2 = χ2√

nmin/npartχ2 independent of the resolution

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AGREEMENT WITH NFW

In cosmological models with no dark energy, halos aremore in agreement with the NFW profile.

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HALO GROUPS

3 HALO GROUPS

I Halos fitted towithin 1σ: 68 %

I Halos fitted towithin 2σ: 95 %

I Halos ill-fitted: 5%

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CONCENTRATION AS A FUNCTION OF MASS

In ill-fitted halos, c has no physical meaning.

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SPARSITY

s∆ =M200

M∆

I no fittingrequired

I probes theouter partsof the halo

∆ =2006001200

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SPARSITY AS A FUNCTION OF MASS

The sparsity is not influenced by the profile agreementwith the NFW profile.

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CONSISTENCY RELATION

near mass independence⇒ Consistency relationbetween s∆ and the mass functions

∫ M2

M1

1M∆

dnd ln M∆

d ln M∆ = s∆

∫ s∆M2

s∆M1

1M200

dnd ln M200

d ln M200

solid:z = 0dotted:z = 1

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VARIATIONS WITH THE COSMOLOGY

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ORIGINS OF THE VARIATIONS

The higher D+ (σ8), the lower s∆

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OBSERVATIONAL DATA

22 clusters with measured M112 and M500 (projected)masses (Local Cluster Substructure Survey)

Okabe et al., 2010

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PROOF-OF-CONCEPT

submitted in Balmès, Rasera, Corasaniti & Alimi, 2013

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CONCLUSIONS

I through Bayesianevidence,selection of anhomogeneoussubsample oflenses, fit forcosmologicalparameterinference

I sparsity offers asimple way to usethe profile of darkmatter halos totest thecosmology

Thank you

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d nfw s irène balmÈs

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