Statistical Astronomy

Transcription

1 Lectures for the 7 th IAU ISYA Ifrane, nd 3 rd Juy 4 p ( x y, I) p( y x, I) p( x, I) p( y, I) Statistica Astronomy Martin Hendry, Dept of Physics and Astronomy University of Gasgow, UK

2 Parameter estimation: the Gaussian approximation Posterior Likeihood Prior p ( mode data, I ) p (data mode, I ) p (mode I )

3 Parameter estimation: the Gaussian approximation p ( data, I ) p (data, I ) p ( I ) Best estimator: ( data, ) p I Equivaenty, we can define Tayor expand ( ) around : Maximise posterior ikeihood og p( data, I ) and compute + ( ) ( ) + ( ) + ( ) K

4 Parameter estimation: the Gaussian approximation p ( data, I ) exp Negecting higher order terms in where A [ ( )] ( ) A p ( data, I ) exp This is equivaent to a norma distribution, with Can summarise inference from posterior by ± σ Gaussian approximation ( ) σ A

5 Parameter estimation: -D case Reca our definition of variance var [] ( x x ) x p( x I) dx Extend to variabes covariance cov [ x, y] ( x x )( y y ) p( x, y I) dxdy x y If and are independent, [ x, y] cov This is because p ( x, y I) p( x I) p( y I)

6 Parameter estimation: -D case p (, data, I ) p (data,, I ) p (, I ) Best estimator: p (, data, I ) Compute where (, data, I ) og p

7 Tayor expand around : ), ( ( ) [ ], exp ) data,, ( I p ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) K ,, Parameter estimation: -D case [ ] Q exp Gaussian approximation χ Maximising ikeihood Minimising χ

8 Tayor expand around : where This is a bivariate norma distribution with covariance matrix ), ( ( ) B C C A Q Quadratic form C B A Parameter estimation: -D case ( )( ) cov i i i i i σ Fisher information matrix

9 Parameter estimation: -D case p(, data, I) Minimum χ maximum This is a bivariate norma distribution with covariance matrix σ i cov i ( )( ) i i Fisher information matrix i

10 Parameter estimation: -D case p(, data, I) Minimum χ maximum sice of χ max This is a bivariate norma distribution with covariance matrix σ i cov i ( )( ) i i Fisher information matrix i

11 Parameter estimation: -D case p(, data, I) Minimum χ maximum sice of χ max This is a bivariate norma distribution with covariance matrix σ i cov i ( )( ) i i Fisher information matrix i

12 Parameter estimation: -D case p(, data, I) Minimum χ maximum sice of χ max This is a bivariate norma distribution with covariance matrix σ i cov i ( )( ) i i Fisher information matrix i

13 Parameter estimation: -D case p(, data, I) We can compute the χ that corresponds to e.g. 68%, 95%, 99% of the posterior pdf. We can draw contours of equa probabiity Confidence regions for the parameters Extends easiy to N parameters or degrees of freedom From Numerica Recipes

14 Parameter estimation: -D case p(, data, I) Contours of constant probabiity are eipses. Covariance matrix is not in genera diagona What we infer about and is not independent

15 Parameter estimation: -D case ρ. ρ. 3 Can define correation coefficient [, ] [ ] var[ ] cov ρ ρ var Covariance matrix becomes ess diagona ρ.5 ρ. 7 ρ increases isoprobabiity contours eongate.7 ρ ρ. 9 Very important if we are interested ony in one parameter

16 Parameter estimation: -D case σ σ ( I ) Best-fit vaue of, found from (, I ) σ σ

17 Parameter estimation: -D case σ σ ( I ) Best-fit vaue of, found from If we ignore the covariance, we seriousy underestimate the uncertainty on (, I ) σ σ

18 Parameter estimation: -D case From Numerica Recipes

19 Parameter estimation: -D case σ σ ( I ) Best-fit vaue of, found from (, I ) σ σ Margina and conditiona error bars ony equa if cov[, ]

20 Parameter estimation: -D case Linear combination of and we constrained by data Length of axes determined by the eigenvaues of the Fisher information matrix Fi [ σ ] i F i λ F determines how much information we can earn about our parameters Direction of axes are the eigenvectors of F

21 Tayor expand around : ), ( ( ) [ ], exp ) data,, ( I p ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) K ,, Parameter estimation: Gaussian approximation [ ] Q exp Gaussian approximation Is the Gaussian approximation a good idea?

22 Tayor expand around : ), ( ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) K ,, Parameter estimation: Gaussian approximation Is the Gaussian approximation a good idea? o Greaty simpifies cacuations ony need to compute the eements of the Fisher matrix (covariance matrix) o Nowadays much better to compute fu posterior pdf. Not too hard with present-day computers, even for arge N Markov Chain Monte Caro Methods

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24 Λ CDM From Lineweaver (998)

25 Genera Reativity:- Geometry matter / energy Spacetime tes matter how to move and matter tes spacetime how to curve Einstein s Fied Equations µν µν µν G R g R 8 π GT µν Einstein tensor Ricci tensor Metric tensor Curvature scaar Energy-momentum tensor of gravitating mass-energy

26 Genera Reativity:- Geometry matter / energy Spacetime tes matter how to move and matter tes spacetime how to curve Einstein s Fied Equations g µν Given can compute and ; These are generated by R T µν µν R

27 Treat Universe as a perfect fuid µν µ ν T ( ρ + P) u u + Pg µν N.B. c Density Pressure Four-veocity Sove to give Friedmann s Equations H R& R 8π Gρ 3 k R R&& R 4π G ρ 3 ( + 3P)

28 Einstein originay sought static soution i.e. :- R & for a t ρ, P R & But if can t have However, GR actuay gives Covariant derivative G µν µν ; ν T ; ν µν µν g Can add a constant times to G µν R µν g µν R + G g µν Λ Einstein s cosmoogica constant

29 Friedmann s Equations now give:- H R& R 8π Gρ + 3 Λ 3 k R R&& R 4 π G ( ρ + 3P) + Λ 3 3 Λ R & for a Can tune to give but unstabe (and Hubbe expansion made idea redundant) But Lambda term coud sti be non-zero anyway! t

30 Cosed Open Fat k curvature constant,, +, open fat cosed

31 Einstein s greatest bunder?

32 Re-expressing Friedmann s Equations:- For Λ H 8π Gρ k 3 R k 8π G ρ ρ 3H crit Define ρ 8π Gρ 3H Ωm Ω ρ Λ Ω k crit Λ 3H k R H It foows that, at any time Ω m + Ω Λ + Ω k

33 Vaue of Ω Ω Λ Ω m R / R Present-day If the Concordance Mode is right, we ive at a specia epoch. Why?

34 Hubbe diagram of distant supernovae Consider an obect of intrinsic uminosity from which we observe a fux I Define the Luminosity Distance via:- L I L 4π d L Distance required to give observed fux if Universe has a fat geometry Actua distance depends on true geometry, and expansion history of the Universe d L d L ( z; Ωm, ΩΛ ) ( + z) dang ( z; Ω m, Ω Λ )

35 Adapted from Schmidt () Distance Moduus d L ( m M ) mag 5og + 5 Mpc Fractiona distance change ½(mag change) e.g.. mag difference is 5% distance difference

36 Hubbe diagram of distant supernovae We need a good standard cande, to probe the geometry of the Universe Type Ia Supernovae o o o o Visibe to huge distances Sma scatter in uminosity at peak brightness Observationa programs to detect dozens (s) Can aso use `Light Curve Shape to improve distance estimates

37 Hubbe diagram of distant Type Ia supernovae Modes with different matter density Mode with positive cosmoogica constant Mode with zero cosmoogica constant Straight ine reation nearby Permutter (998) resuts og z

38 At ow redshift, SNa essentiay measure the deceeration parameter SNIa at z.5 q R t R & e. g., q i ( ) R( t) ( t ) Ω i ( + 3w ) Ω m i Ω Λ Adapted from Schmidt ()

39 At ow redshift, SNa essentiay measure the deceeration parameter SNIa at z. q R t R & e. g., q i ( ) R( t) ( t ) Ω i ( + 3w ) Ω m i Ω Λ Adapted from Schmidt ()

40 From Bennett et a (3)

41 Position of first peak sensitive probe of the geometry of the Universe Ω m + Ω Λ + Ω k

42 Tegmark et a (998) SNIa measure:- q Ωm Ω Λ CMBR measures:- Ωk Ω m Ω Λ Together, can constrain:- Ω,Ω m Λ

43 Can we distinguish a constant Λ term from quintessence? Not from current groundbased SN observations (combined with e.g. LSS) or from future groundbased observations Adapted from Schmidt ()

44 Can we distinguish a constant Λ term from quintessence? Not from current groundbased SN observations (combined with e.g. LSS) or from future groundbased observations Adapted from Schmidt ()

45 Can we distinguish a constant Λ term from quintessence? Not from current groundbased SN observations (combined with e.g. LSS) or from future groundbased observations Main goa of the SNAP sateite (aunch ~?) Adapted from Schmidt ()