d nfw s irène balmÈs
TRANSCRIPT
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
DEUS
NFW PROFILE
SPARSITY
DARK MATTER HALOS
Result of the growthof initialperturbationsinfluenced bygravity andexpansion.
Traced byobservablegalaxies, lensingeffect,...
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DARK MATTER HALOS
DEUS
NFW PROFILE
SPARSITY
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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DARK MATTER HALOS
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WHAT IS A GRAVITATIONAL LENS?
DENOMINATION
2 images : double lens4 images : quadruple lens
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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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DARK MATTER HALOS
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TIME DELAY SURFACE
source plane
lens plane
observer view
time arrival surface
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TIME DELAY SURFACE
source plane
lens plane
observer view
time arrival surface
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TIME DELAY SURFACE
source plane
lens plane
observer view
time arrival surface∆t
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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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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
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RECOVERED VALUE OF H0Starting value for h = H0/100 km/s/Mpc: 0.72
ANALYSIS OF S1 AND S0 + S1 WITHM0
Results heavily biased towards high values of h.
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RECOVERED VALUE OF H0Starting value for h = H0/100 km/s/Mpc: 0.72
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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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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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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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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SPARSITY
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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TIME-DELAYS ANDSPARSITY
IRÈNE BALMÈS
TIME-DELAYS
TIME-DELAYS
BAYESIAN ANALYSIS
ANGULAR STRUCTURE
TEST
RESULTS
SPARSITY
DARK MATTER HALOS
DEUS
NFW PROFILE
SPARSITY
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
46 / 46