Adam S. Trotter UNC-Chapel Hill, Dept. of Physics & Astronomy PhD Final Oral Examination

1. The Gamma-Ray Burst Afterglow Modeling Project (AMP):Foundational Statistics and Absorption &Extinction Models Adam S. Trotter UNC-Chapel Hill, Dept. of Physics & Astronomy PhD Final Oral Examination 30 June 2011 Advisor: Prof. Daniel E. Reichart

2. AMP: The GRB Afterglow Modeling Project Model, in a statistically sound and self-consistent way, every GRB afterglow observed since the first detection in 1997, using all available radio, infrared, optical, ultraviolet and X-raydata. Can get physical information about GRBs… Eiso, εe, εB, p, jet geometry Can get physical information about GRB environments… n(r)rk, AV, NH, Extinction Curves, Dust/Gas Modification

3. Forge a Tool: Statistic • A new statistical technique for fitting models to 2D data with uncertainties in both dimensions • Build an Instrument: Model • GRB emission • MW extinction • Source-frame extinction & absorption • IGM absorption Test it Out: Fit the First GRB 090313 z = 3.375 IR/optical/X-ray data Tests all aspects of model An “Instrumentation Thesis”

4. Forge a Tool: The TRF Statistic A new statistical formalism for fitting model distributions to 2D data sets with intrinsic uncertainty (error bars) in both dimensions, and with extrinsic uncertainty (slop) greater than can be attributed to measurement errors alone

5. syn sy sx sxn The General Statistical Problem: Given a set of points (xn,yn) with measurement errors (sxn,syn), how well does the model distribution fit the data? Joint Probability of Model Distribution and Data Model Distribution = Curve yc(x) convolved with 2D Gaussian yc(x) So, how do we compute pn?

6. sy sx syn sxn (xn , yn) yc(x) It can be shown that the joint probability pn of these two 2D distributions is equivalent to...

7. ...a 2D convolution of a single 2D Gaussian with a delta function curve: Syn Sxn (xn , yn) yc(x) But...the result depends on the choice of convolution integration variables. Also...the convolution integrals are not analytic unless yc(x) is a straightline.

8. If yc(x) varies slowly over (Sxn, Syn), we can approximate it as a line ytn(x) tangent to the curve and the convolved error ellipse, with slope mtn= tanqtn (xtn, ytn) Syn qtn Sxn (xn , yn) yc(x) ytn(x)

9. Now, we must choose integration variables for the 2D convolution integral Syn Sxn (xn , yn) yc(x) y¢tn(x)

10. Both D05 and R01 work in some cases, and fail in others... A new duis needed.

11. A New Statistic: TRF

12. Starting to look like 2… same for all statistics

13. A New Statistic: TRF

14. A New Statistic: TRF

15. D05 TRF R01

16. A New Statistic: TRF Analytically Invertible: same fit to y vs. x as x vs. y Reduces to 2 in 1D limits

17. 2-like measured in direction of closest approach of curve to data point…intuitive!

18. y x Circular Gaussian Random Cloud of Points TRF D05

19. x y y x Circular Gaussian Random Cloud of Points TRF D05

20. y x Circular Gaussian Random Cloud of Points D05 1/myx TRF mxy= 1/myx D05 mxy0

21. Expected Fits to an Ensemble of Gaussian Random Clouds D05 p(q )µ cosNq Strongly biased towards horizontal fits TRF p(q )= const No direction is preferred over another

22. Actual Fits to Ensemble of 1000 Gaussian Random Clouds

23. But…TRF is not Scalable s cannot be factored out of total joint probability Best-fit curve depends on choice of s Distribution of slop into x-and y-dimensions depends on s

24. TRF at s = Inverted D05 TRF at intermediate s TRF at s0 = D05 (This is what Excelwould give) Slop-Dominated Linear Fit

25. Inverted D05 TRF at intermediate s D05 sminis scale where fitted slop σx0 D05 limited to inversion or non-inversion TRF can fit to a continuum of scales TRF at smin TRF at smax smaxis scale where fitted slop σy0 Linear Fit with Slop and Error Bars

26. Range of Physical Scales 0 as Error Bars Dominate Over Slop

27. s0 smax Pearson Correlation Coefficient R2xy = myxmxy Useless for Invertible Statistic R2xy1 smin TRF Scale-Based Correlation Coefficient used to find “optimum scale” s0

29. s0 smax smin TRF can be generalized to non-linear fits…

30. (See Appendix A, B…) …And to asymmetric intrinsic and extrinsic uncertainties

31. Build an Instrument: Models GRB emission MW Dust Extinction Source-Frame Dust Extinction Source-Frame Lyα Absorption IGM Lyaforest/Gunn-Peterson Trough

32. Anatomy of GRB Emission Burst r ~ 1012-13 cm tobs < seconds Afterglow r ~ 1017-18 cm tobs ~ minutes - days Piran, T. Nature422, 268-269.

33. Synchrotron Emission from Forward Shock: Typically Power Laws in Frequency and Time p < -2 log N(E) Em log E See, e.g., Meszaros& Rees, 1997; Sari et al., 1998; Piran, 1999; Chevalier & Li, 1999; Granot et al., 2000; Meszaros, 2002.

34. Sources of Line-of-Sight Absorption and Extinction Circumburst Medium IGM Milky Way Host Galaxy Jet GRB Modified Dust and Gas Host Dust and Gas GP Trough Lya Forest MW Dust

35. Parameters & Priors • The values of some model parameters are known in advance, but with some degree of uncertainty. • If you hold a parameter fixed at a value that later measurements show to be highly improbable, you risk overstating your confidence and drawing radically wrong conclusions from your model fits. • Better to let that parameter be free, but weighted by the prior probability distribution of its value (often Gaussian, but can take any form). • If your model chooses a very unlikely value of the parameter, the fitness is penalized. • As better measurements come available, your adjust your priors, and redo your fits. • The majority of parameters in our model for absorption and extinction are constrained by priors. • Some are priors on the value of a particular parameter in the standard absorption/extinction models (e.g., Milky Way RV). • Others are priors on parameters that describe model distributions fit to correlations of one parameter with another (e.g., if a parameter is linearly correlated with another, the priors are on the slope and intercept of the fitted line).

36. Historical Example: The Hubble Constant Sandage 1976: 55±5

37. Extinction/Absorption Model Parameters & Priors • GRB Host Galaxy: • Prior on zGRB from spectral observations {1} • Assume total absorption blueward of Lyman limit in GRB rest frame • Dust Extinction in Source Frame: • Free Parameters: AV, c2, c4 [3] • Priors on: c1(c2), RV(c2), BH(c2), x0, g from fits to MW, SMC, LMC stellar measurements (Gordon et al. 2003, Valencic et al. 2004) {22} • Damped Lya Absorber: • Free Parameter: NH [1] • Lya Forest/Gunn-Peterson Trough: • Priors on T(zabs) from fits to QSO flux deficits (Songaila 2004, Fan et al. 2006) {6} • Dust Extinction in Milky Way (IR-Optical: CCM model): • Prior on: RV,MW{3} • Prior on: E(B-V)MW from Schlegel et al. (1998) {1} • Total: [4] free parameters, {33} priors

38. IR-UV Dust Extinction Model Cardelli, Clayton & Mathis (1988), Fitzpatrick & Massa (1988) FM Model CCM Model UV Bump Height slope = c2 c1 -RV = -AV / E(B-V)

39. c1 vs. c2 Linear Model Fit to 441 MW, LMC and SMC stars

40. UV Extinction in Typical MW Dust: c2 ~ 1, RV ~ 3

41. Extinction in Young SFR: c2 ~ 0, E(B-V)small, RV large Stellar Winds “Gray Dust”

42. Extinction in Evolved SFR: c2 large, E(B-V)large, RV small SNe Shocks

43. RV vs. c2 Smoothly-broken linear model Fit to 441 MW, LMC and SMC stars Orion SMC

44. The UV Bump • Thought to be due to absorption by graphitic dust grains • Shape is described by a Drude profile, which describes the absorption cross section of a forced-damped harmonic oscillator • The frequency of the bump, x0, and the bump width, g , are not correlated with other extinction parameters, and are parameterized by Gaussian priors. • The bump height, c3 / g 2 , is correlated with c2, with weak bumps found in star-forming regions (young and old), and stronger bumps in the diffuse ISM...

45. Bump Height vs. c2 Smoothly-broken linear model Fit to 441 MW, LMC and SMC stars SMC Orion

46. Gunn-Peterson Trough Lya Forest Absorption Priors Transmission vs. zabs in 64 QSO Spectra

47. Typical GRB Absorption/Extinction Model Spectra

48. Test it Out: Fit the First GRB 090313, z = 3.375 Fit Models to NIR/optical/X-ray Observations Lyα Forest and Lyman Limit in Optical UV Extinction in NIR/Optical Obtain Dust Extinction Curve in High-z SFR …Possibly Modified by GRB Galapagos-Enabled Science: Parameter Linking …Rebrightening: Intrinsic or Extrinsic?

49. GRB 090313: z = 3.375 UV Bump Lyman Limit Lyα Lyα Forest

50. k = 0 k = -2 = p/2= -1.12 c Cooling break mostly below the NIR/optical Cannot distinguish between ISM (k = 0) and Stellar Wind (k = -2) Models