Strong Lens Modeling (I): Principles and Basic Methods

Transcription

1 Strong Lens Modelng (I): Prncples and Basc Methods Chuck Keeton Rutgers, the State Unversty of New Jersey Least-Squares

2 (I) Prncples and Basc Methods least-squares fttng solvng lens equaton constrants (pont data) parametrc mass models (II) Statstcal Methods Bayesan statstcs Monte Carlo Markov Chans nested samplng (III) Advanced Technques case studes: composte models, astrophyscal prors, substructure extended sources non-parametrc lens models Least-Squares

3 Strong lens modelng goal: use strong lensng data to learn about... mass model source other parameters (e.g., H 0 ) focus: galaxy-scale lensng pont data (for now) Least-Squares

4 Smple examples forward problem: fx lens model, solve lens equaton to fnd mage postons (and other data) nverse problem: fx lens data, (re)nterpret lens equaton as constrant equaton solve for model parameters Least-Squares

5 ! #" double lens; conventon: θ 1 > θ 2 > 0 " "!!" β = θ 1 θ2 E θ 1 β = θ 2 θ2 E θ 2 ( β for #2 because mage/source on opposte sdes of lens) ( 1 θ 1 + θ 2 = θe ) θ E = (θ 1 θ 2 ) 1/2 θ 1 θ 2 Least-Squares

6 ! #" " "!!" Least-Squares double lens; agan θ 1 > θ 2 > 0 β = θ 1 θ E β = θ 2 θ E then θ E = θ 1 + θ 2 2 = θ 2

7 Model dependence: Ensten radus remark: from the same data we can get dfferent answers dependng on what we assume about the models however... suppose θ 1 = θ 0 + δ and θ 2 = θ 0 δ, and δ s small: ptmass: θ E = (θ 1 θ 2 ) 1/2 θ 0 δ2 2θ 0 + O ( δ 4) : θ E = θ 1 + θ 2 2 = θ 0 result for Ensten radus s not very senstve to choce of model may not be true of other parameters! Least-Squares

8 lens equaton, now n cartesan angular coordnates [ ] γx u = x θ E ˆx γy cross quad: u = v = 0, wth mages at (±x 1, 0) and (0, ±y 2 ) 0 = (1 γ)x 1 θ E 0 = (1 + γ)y 2 θ E!$#" ' %&!! " #$%& Least-Squares

9 !$#" ' %&!! " #$%& Least-Squares θ E + γx 1 = x 1 θ E γy 2 = y 2 then [ 1 x1 1 y 2 ] [ θe γ soluton ] = [ x1 y 2 ] θ E = 2x 1y 2 x 1 + y 2 and γ = x 1 y 2 x 1 + y 2

10 Least-squares fttng usually we cannot solve the constrant equatons exactly more constrants than parameters nose wrong model general goal: mnmze the dfference between the model and data quantfy goodness of ft: dea: fnd best ft (mnmum χ 2 ) χ 2 = (model data) 2 (uncertantes) 2 explore range of allowed models (regon where χ 2 s acceptable) Least-Squares

11 What s good enough? quantfy degrees of freedom: ν = (# constrants) (# free parameters) f errors are random, have probablty dstrbuton for χ 2 : p(χ 2 ν) = ν/2 Γ(ν/2) (χ2 ) ν/2 1 e χ 2 /2 Least-Squares

12 average: peak: χ 2 = ν χ 2 peak = max(ν 2, 0) as a rule of thumb, we expect χ 2 ν for a good ft; but gven statstcal scatter, ths s not a strct condton! Least-Squares

13 generalze noton of uncertantes... f uncertantes are correlated, ntroduce covarance ( )( ) Cov(x, y) = x x y y = xy x y x y + x y = xy x y for an array of data d = (d 1, d 2, d 3,...), covarance matrx s σ1 2 Cov(d 1, d 2 ) Cov(d 1, d 3 ) Cov(d 2, d 1 ) σ2 2 Cov(d 2, d 3 ) C = Cov(d 3, d 1 ) Cov(d 3, d 2 ) σ Least-Squares

14 5 1 [ C = ] ρ 12 = asde: correlaton coeffcent (dmensonless, ρ 1): ρ j = Cov(d, d j ) σ σ j Least-Squares

15 generalzed goodness of ft χ 2 = (d mod d obs ) t C 1 (d mod d obs ) f data are ndependent then σ1 2 0 C = 0 σ and χ 2 reduces to what you expect χ 2 = = d mod 1 d obs 1 d mod 2 d obs 2. (d mod t d obs ) 2 σ 2 1 σ σ d mod 1 d obs 1 d mod 2 d obs 2. Least-Squares

16 Lnear parameters example: x s some ndependent varable (whch we can know); measure d obs and postulate a straght lne d mod = mx + b Least-Squares

17 χ 2 = (mx + b d obs ) 2 σ 2 parabola n both m and b; fnd mnmum by solvng 0 = χ2 m 0 = χ2 b = 2 = 2 x (mx + b d obs ) σ 2 (mx + b d obs ) σ 2 may look complcated, but just a par of lnear equatons [ x 2 ] x [ ] x d obs σ 2 σ m 2 = σ 2 b x σ 2 solve by matrx nverson 1 σ 2 d obs σ 2 Least-Squares

18 [ x 2 σ 2 x σ x σ 2 1 σ 2 ] [ m b ] = x d obs σ 2 d obs σ 2 (can generalze to an arbtrary number of lnear parameters) Least-Squares

19 Non-lnear parameters must explctly search parameter space use establshed algorthms to search for mnmum of a functon n multple dmensons challenges: computatonal effort local mnma long, narrow valleys degeneraces Least-Squares

20 Downhll smplex method ( amoeba ) brooks/papers/amoeba.pdf also Numercal Recpes Orgnal Smplex Reflecton Expanson Contracton Mult-dmensonal Contracton Least-Squares

21 parameters suppose we have parameters a and b such that then optmal value of a: d mod = a f(b) χ 2 (a, b) = [af(b) d obs ] 2 0 = χ2 a = 2 f(b)[af(b) d obs ] σ 2 a opt = then σ 2 χ 2 (b) = χ 2 (a opt (b), b) we can stll optmze the lnear parameters analytcally f(b)d obs /σ 2 f(b)2 /σ 2 Least-Squares

22 lkelhood 1-d Gaussan χ 2 = (x d)2 σ 2 L e χ2 /2 { ±1σ : χ 2 = 1 (68%) ±2σ : χ 2 = 4 (95%) Least-Squares Σ central regon = 68% of the probablty; each tal = 16%

23 2-d Gaussan f Z 1 x2 y2 = exp 2 2 dx dy 2πσx σy < χ2 2σx 2σy 2 Z Z χ2 2 2 x + y 1 exp dx dy = e r /2 r dr = 2π < χ2 2 0 ( % : χ = 2.3 = 1 e χ /2 95% : χ2 = 6.2 Least-Squares

24 Solvng the lens equaton challenges: usually non-lnear often transcendental we may not even know how many solutons there are! mathematcal theorems bound maxmum number of mages... but we need actual number global caustc structure may be nformatve... but dffcult to fnd and analyze soluton: read lens equaton backwards mappng from mage poston x to unque source poston u(x) = x α(x) tle mage plane map each tle back to source plane number of tles that cover source reveals number of mages tles themselves gve estmates of mage postons Least-Squares

25 Least-Squares

26 Image plane tlng background Cartesan grd basc coverage polar grd centered on each galaxy resolve key regons adaptve subgrddng near crtcal curves Least-Squares

27 Quadrlaterals vs. trangles quadrlaterals can be problematc: ok bad trangles are fne: ok ok Least-Squares

28 trangulaton start wth ponts n a plane connect them wth trangles (Google trangulaton I use quake/trangle.html) Least-Squares

29 Grddng n gravlens/ Least-Squares

30 Magnfcaton and tme delay deflecton magnfcaton α(x) = φ(x) = [ φx [ ] 1 1 φxx φ µ = det xy = [ (1 φ φ xy 1 φ xx )(1 φ yy ) φ 2 ] 1 xy yy specal case of crcular symmetry, α(r): (crcular) µ = tme delay [ ] 1 t(x; u) = t 0 2 x u 2 φ(x) φ y [ 1 α(r) ] 1 [ 1 dα ] 1 r dr ] t 0 = 1 + z l c D l D s D ls Least-Squares

31 pont sources data mage postons fluxes tme delays source parameters poston flux tme scale (extended sources on Thursday) Least-Squares

32 Poston constrants exact poston χ 2 : χ 2 pos = (x mod mages x obs ) t S 1 (x mod x obs ) astrometrc uncertantes: error ellpse wth axes (σ 1, σ 2 ) and poston angle θ σ (East of North) covarance matrx [ ] [ ] σ 2 S = R σ2 2 R t sn θσ cos θ R = σ cos θ σ sn θ σ f symmetrc uncertantes: [ ] σ 2 S = 0 0 σ 2 Least-Squares

33 note: defne source poston assocated wth each observed mage also subtract: δu u obs = x obs α(x obs ) u mod = x mod α(x mod ) = δx [ α(x mod ) α(x obs ) ] µ 1 δx provded that model s decent, such that δx and δu are small then δx µ δu yelds approxmate poston χ 2 : χ 2 pos (u mod u obs ) t µ t S 1 µ (u mod u obs ) Least-Squares

34 advantages: χ 2 pos (u mod u obs don t need to solve lens equaton ) t µ t S 1 µ (u mod u obs ) u mod s a lnear parameter, so optmze t analytcally concerns: where A = u mod = A 1 b µ t S 1 µ b = µ t S 1 µ u obs approxmaton s vald only when resduals are small... but χ 2 pos yelds a large value (.e., bad ft) n ether case snce we do not solve the lens equaton, we cannot check that the model predcts correct number of mages... only worry about models yeldng too many mages Least-Squares

35 Flux constrants χ 2 flux = (F obs µ F src ) 2 σ 2 f, f desred, nclude party by lettng F obs optmal source flux can be found analytcally F src = F obs and µ be sgned µ /σ 2 f, µ2 /σ2 f, f desred, straghtforward to swtch to magntudes m mod = m src 2.5 log µ note: photometrc unts are arbtrary absolute fluxes or magntudes, or relatve values Least-Squares

36 Tme delay constrants predcted tme delay model: t mod τ mod = 1 x mod 2 cosmol: t 0 = 1 + z l c = t 0 τ mod + T 0 u mod 2 φ ( x mod ) D l D s D ls = H 1 0 f(ω M, Ω Λ ; z l, z s ) note: tme zeropont T 0 does not affect dfferental tme delays; but let s make framework general then χ 2 tdel = (t obs t 0 τ mod T 0 ) 2 σ 2 t, Least-Squares

37 χ 2 tdel = (t obs t 0 τ mod T 0 ) 2 σ 2 t, f we have prors on the cosmologcal parameters (ncludng H 0 ) pror t 0,pror ± σ t0 addtonal term optmal values of t 0 and T 0 : (τ mod ) 2 σt, 2 τ mod σ 2 t, + 1 σ 2 t0 χ 2 t0 = (t 0 t 0,pror ) 2 σ 2 t0 τ mod σ 2 t, 1 σ 2 t, [ t0 T 0 ] = τ mod t obs σ 2 t, t obs σt, 2 + t0,pror σ 2 t0 Least-Squares

38 Parametrc mass models postulate: mass dstrbuton can be descrbed by a functon wth a modest number of parameters example: Sngular Isothermal Ellpsod (SIE) pros: κ = b 2[(x x 0 ) 2 + (y y 0 ) 2 /q 2 ] 1/2 easy to fnd best ft and assess qualty (+rotaton) buld n astrophyscal knowledge assumptons and prors good enough for many applcatons cons: can only get out what you put n real galaxes may be more complex Least-Squares

39 Countng # constrants: # parameters: x gal x F t total quad double u src F src x gal q gal q env t 0 total Least-Squares

40 softened power law ellpsod κ = b 2 α 2(s 2 + x 2 + y 2 /q 2 ) 1 α/2 where < 1 steeper than sothermal M(r) r α α = 1 sothermal > 1 shallower than sothermal has many other model classes: pont mass, pseudo-jaffe, de Vaucouleurs, Hernqust, Sersc, NFW, Nuker, exponental dsk,... Least-Squares

41 models can combne multple components to obtan models that are more complcated but stll parametrc for example: stellar component (e.g., Hernqust) dark matter halo (e.g., NFW) (composte models can be as fancy as you want) Least-Squares

42 al effects few lens galaxes are solated they have neghbors, and may be embedded n groups or clusters of galaxes envronments can affect the lght bendng by an amount larger than the measurement uncertantes f neghborng galaxes are far from the lens (compared wth Ensten radus), make Taylor seres expanson φ env = φ 0 + a x + κ c 2 r2 + γ 2 r2 cos 2(θ θ γ ) + σ 4 r3 cos(θ θ σ ) + δ 6 r3 cos 3(θ θ δ ) +... structures along the lne of sght can also affect the lght bendng... more complcated Least-Squares

43 parameter space searchng parameter space may or may not requre a strategc approach... Least-Squares

44 : hands-on exercses... step 1 pck some mass model, then: plot grd plot crtcal curves and caustcs fnd mages Least-Squares

45 : step II I generated some mock lenses; now you try to ft them man lens galaxy s a power law ellpsod I may have vared: mass ellptcty/pa power law ndex envronment: shear/pa, or perturber all generated wth z l = 0.3, z s = 2.0, Ω M = 0.27, Ω Λ = 0.73, and some fxed value of H 0 Least-Squares

46 Sample quads recall: z l = 0.3, z s = 2.0, Ω M = 0.27, Ω Λ = 0.73 what are the model parameters? what s H 0? 2 sampquad1 2 sampquad2 2 sampquad sampquad4 2 sampquad5 2 sampquad Least-Squares

47 Sample doubles recall: z l = 0.3, z s = 2.0, Ω M = 0.27, Ω Λ = 0.73 what are the model parameters? what s H 0? 2 sampdoub1 2 sampdoub2 2 sampdoub sampdoub4 2 sampdoub5 2 sampdoub Least-Squares