The Present and Future of Gamma-Ray Burst Cosmology

Transcription

1 The Present and Future of Gamma-Ray Burst Cosmology July Lawrence Berkeley National Laboratories Andrew S. Friedman (Harvard-CfA) & Joshua S. Bloom (now UC Berkeley)

2 Outline!GRB Cosmography: Motivations!GRB Standard Candles: Energetics!The E p -E γ Relation: Current Status!Cosmology with GRBs!Future Prospects with Swift

3 Long Gamma-Ray Bursts (GRBs)!Prompt γ ray emission for t~ s (Short GRBs t< 2s)!Cosmological " z measured from optical afterglow!host galaxies identified!emission in paired jets (some exceptions?)!true rate ~ observed rate!progenitors = massive stars, Type Ib/c SNe

4 GRB Spectra: Band Model!Smooth broken power Law (Band et al. 1993)! Band Spectrum fits most bright BATSE GRBs (Preece et al. 2000) 4/12 GRB Spectra w/ z Amati et al Ep =Peak energy of νfν spectrum

5 Energy Budget E~ erg!internal Shocks - Prompt γ rays (GRB)!External Shocks - Afterglow: X-ray, Optical, Radio

6 Hubble Diagram Precision Cosmology Concordance Model Knop et al

7 An Old Idea One may wonder whether perhaps GRBs would provide a meaningful independent estimate of Ω Perhaps GRBs will not be able to contribute meaningfully to the myriad of measurements before other methods become more precise, but it is likely that they provide a useful consistency check based on independent objects. Cohen & Piran, 1997 Also see Dermer 1992, Rutledge et. al 1995

8 Motivations: GRBs + SNe Ia!GRBs: very bright; detectable out to z >2; both prompt-emission & afterglow (z~10-20?)!γ-rays penetrate dust!more tractable k-corrections!different evolution!different systematics!swift in space (SNAP: z max ~1.7 classify Ia or Ib/c) GRB standard candles could serve as independent probes of cosmic expansion history, complementary to SNe Ia

9 Immediate Concerns Is high-z cosmology interesting?!z > 1.7 is Ω M -dominated!no training set of nearby GRBs!Few low-z GRBs. High-z GRBs complement lower-z SNe Ia!GRB coverage in 1<z<2 already comparable to high-z SNe Ia from HST (Knop et al. 2003, Riess et al. 2004)!SNe Ia survey up to at least z ~ 2 crucial to pin down dark energy systematics (Linder & Huterer 2003)

10 Outline!GRB Cosmography: Motivations!GRB Standard Candles: Energetics!The E p -E γ Relation: Current Status!Cosmology with GRBs!Future Prospects with Swift

11 GRB Energetics (Isotropic Equivalent)!E iso distribution spans several orders of mag. E iso is a bad standard candle Fluence in observed bandpass Cosmological k-correction Luminosity Distance (theory) Hubble constant /70 kms -1 Mpc -1 Data from: Friedman & Bloom 2005 Redshift

12 GRB Jets & Beaming!Beaming fraction f b inferred from achromatic break in afterglow light curve (t~t jet ) Top Hat Jet Model Stanek et. al 1999

13 Beaming Corrected Energy (E γ )!E γ distribution is much narrower than E iso (Frail et al. 2001, Piran et al. 2001, Bloom et al. 2003)!" the Frail Relation : E γ ~ constant 15 GRBs (f b ), 17 GRBs (z) Frail et al GRBs (f b ), 29 GRBs (z) Bloom et al. 2003

14 GRB Energetics & Hubble Diagram!No low-z calibration set " re-fit for standard candle energy for each cosmology!scatter similar for each cosmology!grb standard candle E γ promising but not sufficient for cosmography Bloom, Frail, Kulkarni 2003

15 Standard Energy No Longer!E γ getting worse for new data: e.g , , X-Ray Flash (XRF) , !GRB (z=0.0085) is not the only outlier Frail et al Bloom et al Ghirlanda et al Some still trying to do cosmology w/ Eγ (Di Girolamo et al. 2005)

16 Outline!GRB Cosmography: Motivations!GRB Standard Candles: Energetics!The E p -E γ Relation: Current Status!Cosmology with GRBs!Future Prospects with Swift

17 The E p -E iso Amati Relation Amati et al GRBs w/ z Much discussed correlation: Lamb et al Extends to XRFs E p ~ (E iso ) 1/2

18 The E p -E γ Ghirlanda relation Ghirlanda et. al 2004a E p ~(E γ ) 2/3!Smaller scatter than Amati relation.!based on more physical beaming corrected energy!new standard candle contender?!but E γ depends on cosmology.

19 The Ep-Eγ relation!only ~20 GRBs can be placed on relation χ 2 /dof = 3.71 (17 dof)!poor fit to a power law, at least under our assumptions Friedman & Bloom 2005

20 Measuring the Necessary Observables To place a GRB on the E p -E γ relation, one needs to assume a cosmology, a k-cor bandpass, pick a model for the jet structure (i.e. E γ ), and measure the following (must at least have S γ, z, E p, & t jet ) : Fluence in observed bandpass (S γ ) Peak of prompt-burst spectrum (E p ) Spectroscopic redshift (z) Time of afterglow jet-break (t jet ) Ambient Density (n=10 cm -3?) γ-ray conversion efficiency (ξ=20%) VERY EASY HIT & MISS HARD HARD VERY HARD UNKNOWN

21 Input Assumptions for Eγ! Some parameters unknown for most bursts in the sample; we assume some fiducial value for all these GRBs! Ambient Density n=10 ± 5 cm -3 (50% error)! γ-ray conversion efficiency ξ=20%

22 Goodness of fit sensitive to: 1. n 2. σ n /n 3. Data set 4. References for individual GRBs 5. k-cor band (small # stats) Sensitivity to Input Assumptions χ 2 ν = 3.71: upper left; triangle (Friedman & Bloom 2005) χ 2 ν = 1.27: lower right; diamond (Ghirlanda et al. 2004c) n~1-2 cm -3 optimizes χ 2 ν Friedman & Bloom 2005b

23 Avoiding Sensitivity to Density, Efficiency Recent Work with Empirical Standard Candle with no Jet Model Assumptions: E iso ~ E pα t jet β (Liang & Zhang 2005, Xu 2005 α ~ 2, β ~ -1) Fluence in observed bandpass (S γ ) Peak of prompt-burst spectrum (E p ) Spectroscopic redshift (z) Time of afterglow jet-break (t jet ) Ambient Density (n=10 cm -3?) γ-ray conversion efficiency (η γ =20%) VERY EASY HIT & MISS HARD HARD VERY HARD UNKNOWN

24 Systematic Errors!Cosmological k-correction bandpass choice!neglecting Covariance!Gravitational Lensing!WIND vs. ISM!Assuming the same density for all bursts!assuming the same efficiency for all bursts!selection Effects Nakar & Piran 2004, Band & Preece 2005

25 Outline!GRB Cosmography: Motivations!GRB Standard Candles: Energetics!The E p -E γ Relation: Current Status!Cosmology with GRBs!Future Prospects with Swift

26 GRB Hubble Diagram: Formalism For self-consistency, must re-fit (or marginalize over) slope (η), intercept (κ) of E p -E γ relation in each cosmology. GRB Luminosity Distance [cm] (Small angle approx.) GRB Distance Modulus [mag] Empirical Correction From E p -E γ relation [mag]

27 Improvement in GRB Hubble Diagram E p -E γ E γ!compute χ 2 ν for DM-z relation for each cosmology!(χ 2 ν) min gives favored (Ω M, Ω Λ ) cosmology.!but χ 2 ν ~ 3 for standard (Ω M =0.3, Ω Λ =0.7) cosmology under our assumptions Friedman & Bloom 2005 Is χ 2 ν acceptable for any cosmology?

28 Cosmography with GRBs alone!compute χ 2 ν for each cosmology in grid!(χ 2 ν) min " best fit loitering cosmology (Ω M, Ω Λ ) = (0.12, 1.32) Friedman & Bloom 2005 But!Lack of low-z sample!(χ 2 ν) min > 2 (poor fit) for range of assumptions for n, (σ n /n)!increasing (σ n /n) arbitrarily trivially gives good fit

29 Alternative Cosmological Methods Without nearby training set, poor low-z coverage, must modify traditional Hubble Diagram χ 2 method 1) η, κ re-fit for each cosmology (Ghirlanda et al. 2004a, Friedman & Bloom 2005, Xu, Liang, & Dai 2005 Method I) 2) η, κ marginalized over (Xu, Liang, & Dai 2005 Method II, Mortsell & Sollerman 2005) 3) η, κ fixed (future low-z training set or theory) (Dai et al 2004, Xu, Liang, & Dai 2005 Method III) 4) Weight cosmologies w/ good E p -E γ fits (Ghirlanda et al 2004b minimum scatter, Firmani et al Bayesian, Xu 2005)

30 Dai et al. 2004, Ghirlanda et al. 2004b Dai et. al 2004! Report Ω M =0.35 ± 0.15 assuming flatness! Also try to constrain w! Assumed η=2/3 for all cosmologies " circular without low-z calibration Ghirlanda et al. 2004b!Joint fit w/ SNe Ia, claim GRBs + SNe Ia more consistent w/ WMAP!Fit dominated by SNe!GRBs alone favor loitering model!grb data pulls SNe Ia data in a favorable direction; what if SNe Ia, LSS, & CMB didn t exist?

31 Different Cosmographic Methods 17 GRBs Method I Method II -- Method III 15 GRBs Method IV Xu, Dai, & Liang 2005 Firmani et al Method I: η, κ re-fit for each cosmology Method II: η, κ marginalized over Method III: η, κ fixed (theory or future low-z training set) Method IV: weight cosmologies w/ good E p -E γ fits (Bayesian)

32 Outline!GRB Cosmography: Motivations!GRB Standard Candles: Energetics!The E p -E γ Relation: Current Status!Cosmology with GRBs!Future Prospects with Swift

33 !50+ GRBs localized New Swift Bursts!~10 ground based follow up redshifts!z, E p, t jet not coming from on-board instrumentation.!only 2 post-swift bursts, , a can even be placed on the Ghirlanda relation, and these were coobserved by other satellites (HETE-2, Integral).!050408, a consistent w/ relation There was some hope that Swift would become a GRB cosmology satellite. Unfortunately this is not going to happen. Nature is not cooperating.

34 Simulations of Swift Data Method I: re-fit η, κ for each cosmology (their fig 5) Method II: marginalize over η, κ over a wide range (their fig 6) Red 157 GRBs (simulated) Blue 17 GRBs Xu, Dai, & Liang 2005 Method III: assumes η, κ fixed by low-z sample (their fig 7) Dashed 17 GRBs Red 157 GRBs (simulated) Blue 156 SNe Ia (Gold)

35 Simulations of Swift Data Bromm & Loeb 2002 Redshift distribution Flatness prior Mörtsell & Sollerman 2005 simulate sample of 200 Swift GRBs (Friedman & Bloom 2005) ~650 SNe Ia (Gold + SN Factory + ESSENCE) using SNOC package (Goobar et al. 2002) Bad News!Without low-z sample (unlikely w/ Swift)!GRBs alone sensitive to Ω M, not Ω Λ!GRBs insensitive to (w=w o )or w(t) Good News!LSS priors on Ω M help SNe Ia, but similarly might be achieved w/ only standard candle techniques (GRB prior on Ω M )

36 Conclusions Good!GRBs may constrain Ω M, z T best in future!complementary redshift distribution to SNe Ia!Empirical approach avoids sensitivity to n, ξ Bad!No training set " can t do Ω Λ, w, w(t)!lack of low-z coverage " loitering problem!systematics, selection effects!lack of basis from physics!sample may only double in ~2 years!need Swift + HETE-2 + Integral +???!Even w/ training set, evolution degenerate w/ w(t)

37 Praise for GRB Cosmology Enthusiasts Ghirlanda et al. 2004b Xu, Dai, & Liang 2005 Firmani et al Lazzati et al Ghisellini et al Di Girolamo et al Liang & Zhang 2005 Xu 2005 Lamb et al Cautionists Friedman & Bloom 2005 Mortsell & Sollerman 2005 Bertolami et al 2005

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39 Cosmology-Independent Calibration 1. Low redshift calibration unlikely with Swift!3/39 GRBs detected w/ z <0.2!030329: large outlier in Eγ but falls directly on Ep-Eγ relation low-z anchor of the relation (z=0.1685)!however, 2 lowest z GRBs (980425, ) are outliers independent of density assumptions 2. Could know slope (η) of relation a priori from physics, obviating need to re-fit in every cosmology.!physical basis is not understood: Eichler & Levinson 2004, Rees & Meszaros 2004!Selection effects Band & Preece 2005!On much less stable footing than SNe Ia mag corrections

40 GRBs: Not Ready For Cosmography! Sensitivity to input assumptions! Small number statistics! Few low-z bursts (Ω Λ,w)! Lack of low-z training set to calibrate relation (η), in a cosmology-independent way! χ 2 contours (Ω M,Ω Λ ) may not be meaningful! Current data do not rule out that the Ghirlanda relation will be well fit by a power law, but claims of the cosmographic utility of GRBs require added caution

41 Cosmographic Potential!If..with refinements to relation, expanded data set, GRBs prove to be standardizable candles:!test cosmology to z > 10 (z max ~1.7 for SNe Ia)!Determine Ω M to higher precision!determine transition z T to epoch of deceleration!help pin down dark energy systematics, w(t)!+ No dust/reddening, simple k-corrections, several years before SNAP launches (> 2010?)

42 The Loitering Problem Ω Λ χ 2 ν= 0.98 (min) χ 2 ν= 1.00 (conc) χ 2 ν= 1.07 (flat) 68.3% 90% 95.4% 99% 99.73%!We ran a simulated sample provided by E. Mörtsell through our analysis method re-fitting Ghirlanda relation for each cosmology (vs. marginalizing over η, κ) Ω M Flat!Sample has an unrealistic uniform redshift distribution between z=0.25 and z=2!loitering problem is a reflection of the redshift distribution of the sample, not a particular analysis method

43 Other Potential Applications!If correlation is upheld by future data, one can assume a cosmology and learn about GRBs!Physical basis of correlation may lend insight into GRB radiation mechanisms, jet models!outliers to the relation may indicate separate subclasses of GRBs that may come from different progenitor systems (e.g , ), or at least tell us about GRB diversity! Required follow up observations of independent interest for constraining progenitors, WIND, etc

44 References Amati et al. 2002, A&A, 390, 81 Bloom et al. 2003, ApJ, 594, 674 Dai et al. 2004, ApJ, 612, L101 Frail et al. 2001, ApJ, 562, L55 Friedman, A.S. & Bloom, J.S., ApJ, 627, 1 Ghirlanda et al. 2004a, ApJ, 616, 331G Riess et al. 2004, ApJ, 607, 665 Ghirlanda et al. 2004b, ApJ, 613, L13 Schaefer, B.E. 2003, ApJ, 588, 387 Takahashi et al. 2003, (astro-ph/ ) cosmicbooms.net

45 References Ghirlanda et al. 2004b Xu, Dai, & Liang 2005 Firmani et al Lazzati et al Ghisellini et al Di Girolamo et al (E γ ) Bertolamo et al 2005 Liang & Zhang 2005 Xu 2005 Mortsell & Sollerman cosmicbooms.net

46 Gamma-Ray Bursts: Cosmological?!Isotropy " Strong Evidence For Cosmological Origin

47 Long GRBs: Progenitors? Collapsar Model!Core Collapse Type Ib/c SNe - Massive (~20-40M ) Wolf-Rayet Star (H, He envelope lost) " BH!Stellar Material Forms Accretion Disk Around Rapidly Rotating BH!Accretion Lifetime ~ Duration of GRB (~ s)!relativistic Jets (How?)!Energy Sources: Potential Energy From Disk Mass + BH Spin (B-Z)!E~ erg McFadyen, Woosley, & Heger 2001

48 Bimodal Duration Distribution (Kouveliotou et. al 1993) Duration GRBs: T 90 < 2s (~25%)!Long Duration GRBs: T 90 > 2s 1000s (~75%)!Bimodal Dist. " 2 Distinct Progenitor Subclasses

49 Afterglow Era: present!bepposax: X-Ray Afterglow for GRB detected (Costa et. al 1997)!Optical Follow-Up (vanparadijs et. al 1997, Frail et. al 1997)!GRB , first redshift detection (z = 0.835) (Metzger et. al 1997)!GRB Host Galaxy Identification (Bloom et. al 1998)!Case for cosmological origin of at least some long GRBs is sealed

50 Fireball Model Rees & Meszaros, 1992, 1994!Generic even without knowledge of central engine

51 Association w/ Star Forming Regions!Offset distribution from host galaxy center does not fit predicted distrib. for binary merger (e.g. NS-NS) models Bloom et. al 2002!Fits offset distribution associated with star forming regions " favors massive star progenitors for long GRBs.

52 GRB-Supernova Connection!GRB /SN 1998bw (Eiso ~ erg)!grb : Late Bump in Light Curve (Bloom et. al 1999) Also some evidence for bumps in ~10 other GRBs (Bloom et. al 2003, IAU Colloquium)! /SN 2001ke (late-time bump) (Bloom et. al 2002, Garnavich et. al 2003)!GRB /SN 2003dh First Spectroscopic Confirmation (Stanek et. al 2003)! /SN 2002lt (della Valle et. al 2004)!GRB /SN 2003lw

53 Motivations!GRBs: brightest explosions in universe!detectable out to high z >5 (easy), z max ~10-20 (?)!Gamma-rays penetrate dust / cleaner k-corrections!any evolution likely orthogonal to Type Ia SNe!Swift satellite in space! SNAP > 2010? (z max ~ 1.7) A GRB standard candle could serve as an independent probe of the geometry & expansion history of the universe, complementary to SNe Ia

54 Can Different Jets Save Eγ?! Alternative Jet Models (Structured Jets Not Top Hats): *Power Law: Rossi et. al 02, Zhang & Meszaros 2002 *Gaussian Jet: Zhang & Meszaros 2004, Lloyd- Ronning et. al 2004! Perhaps the low Eγ GRBs are viewed off-axis?! Structured jets w/ more energy on-axis find natural support in numerical simulations of collapsar model: (MacFadyen et. al 1999, 2001)! But alternative jet models come at expense of new, unknown free parameters, which may vary between GRBs!Even these can not recover standard Eγ for , XRFs! Give up? Or combine with other correlations?

55 GRB Redshift Indicators!Previous GRB redshift indicators used prompt γ ray properties alone, using correlations found between isotropic luminosity (Liso) and!variability: Fenimore & Ramirez Ruiz 2000, Reichardt et. al 2001, Lloyd-Ronning & Ramirez-Ruiz!Spectral Lags: Norris et. al 2000, Noriss 2002, Schaefer et. al 2001!Renewed enthusiasm for cosmographic utility of GRBs: Schaefer 2003, Takahashi 2003!Focus on those derived from energetics

56 Cosmology Dependence Friedman & Bloom 2005 (Asterisk)!Slope does not vary dramatically between cosmologies!however, it must be re-fit in each one for self-consistent cosmography

57 Sensitivity to Input Assumptions

58 Improvement Major reduction in scatter from Eiso, Eγ as GRB standard candles This Talk 3

59 GRB Hubble Diagram!Goodness of fit (χ 2 /dof) for distance modulus-relation computed for each cosmology in grid.!(χ 2 /dof) min gives favored (Ω M, Ω Λ ) cosmology.!but χ 2 /dof> 3 under our assumptions So is χ 2 /dof acceptable for any (Ω M, Ω Λ )?

60 Motivations Also see Dermer 1992, Rutledge et. al 1995

61 GRB Standard Candles? What function of observables might serve as a useful GRB standard candle? Purely empirical approach.! It must be roughly constant from burst to burst or at least come from a narrow intrinsic distribution! It must not evolve (or evolve weakly) with redshift! It can depend on properties of both the prompt emission and the afterglow, although the latter are more observationally expensive! Search for a standard candle, some function of GRB observables in the set of ~40 bursts with known z! Focus on energetics, although others proposed

62 Analysis of Dai et. al 2004 cont!differences stem from input assumptions, data selection, and method of analysis!dai et. al exclude the two largest outliers: , ( other bursts)!assumed n=3 cm -3, greatly improving the fit compared to our assumption (n=10 cm -3 )!Assumed slope of relation fixed (η=2/3) for all cosmologies. This slope was fit assuming the WMAP cosmology. Circularity problem.

63 ! Present joint fit w/ SNe Ia, claim GRBs + SNe Ia more consistent with WMAP! Some improvements: re-fit η in each cosmo. Ghirlanda et. al 2004b! But it was not possible to reproduce their results from their papers alone! Few equations! Assumptions unclear! χ 2 /dof not reported for Ep-Eγ relation Ghirlanda et. al 2004b

64 Analysis of Ghirlanda et. al 2004b!Improvements from Dai. et. al 2004!Self-consistent re-calibration of relation (η) for each cosmology!stressed need for low-z calibration bursts!ghirlanda et. al fit to Ep-Eγ relation (χ 2 =1.27) compared to our (χ 2 =3.71), highlights extreme sensitivity to input assumptions, data selection!assumptions not made clear in their paper!not possible to reproduce their results without further investigation (conference + conversations)!investigation elucidated sensitivity to assumptions

65 GRBs + SNe Ia: independent objects, techniques, different systematics

66 Sensitivity to Input Assumptions! 2 lowest z GRBs " outliers independent of assumptions (z=0.0085) (z=0.1055) Bad for future training set!! Essentially all other nominal outliers can be made to fall on relation by changing density (efficiency), errors Friedman & Bloom 2005 ApJ, 627, 1, July 2005! There exist assumptions, data subsets with good fits, (e.g. Dai et. al, Ghirlanda et. al), but these are not necessarily favored a priori

67 Updated Comparison: Eγ vs. Eiso 39 GRBs (z) 33 GRBs (f b ), 39 GRBs (z) Data from: Friedman & Bloom 2005 ApJ, 627, 1, July 2005

68 Controversy!At GRBs in the Afterglow Era: 4 th Workshop, Rome Italy, Oct 18-22, 2004, Ghirlanda et. al report at good fit (χ 2 ν = χ2 /dof = 1.27; 13 dof) to the Ep-Eγ relation!can we reconcile this with our poor fit? (χ 2 ν = 3.71; 17 dof), by comparing assumptions, data sets?

69 Firmani et. al 2005!Tries to solve loitering problem (resulting from largely high-z sample) for existing data with a Bayesian approach that give extra weight to the good cosmologies!does not take into account sensitivity to input assumptions, i.e. density, efficiency Red Blue Blue Dashed Colored Dashed 15 GRBs 156 SNe Ia (Gold) 14 SNe Ia z>0.9 Joint Confidence Interval transition redshift, matter vs. Λ

70 Firmani et. al 2005 Red 15 GRBs Blue 156 SNe Ia (Gold) Blue Dashed 14 SNe Ia z>0.9 Colored Joint Confidence Interval!Bayesian method gives more favorable contours for current data than traditional χ 2 statistics

71 Ghirlanda et. al 2004b! Joint fit w/ SNe Ia, claim GRBs + SNe Ia more consistent with WMAP!Fit dominated by SNe!GRBs alone: almost inconsistent w/ Big Bang (i.e. loitering cosmology)!grb data pulls SNe Ia data in a favorable direction!but what if SNe Ia, LSS, & CMB didn t exist? Ghirlanda et al. 2004b Must rigorously test GRBs alone before combining with SNe Ia

72 Mörtsell & Sollerman 2005!Simulate sample of 200 GRBs with properties similar to Friedman & Bloom sample (~20 GRBs)!Instead of re-fitting for slope, intercept, of Ghirlanda relation for each cosmology, Mörtsell & Sollerman 2005 marginalize over them (analogous to M for SNe Ia), similar results as long as one does not assume the slope to be fixed!simulate sample of ~650 SNe Ia w/ SNOC package Goobar et al (Gold + ESSENCE + SN Factory) With few bursts available at low z, uncertainty in redshift distribution of future Swift sample, GRBs alone sensitive to Ω M, not dark energy Ω Λ, its equation of state w, or its possible time variation.

73 Mörtsell & Sollerman 2005!Without low-z training set, GRBs alone sensitive to Ω M, not Ω Λ!LSS priors on Ω M (2dF: Percival et al. 2001, SDSS: Tegmark et al. 2004) help SNeIaw/ or w/o flatness!same or better might be achieved by assuming a GRB prior on Ω M standard candle techniques alone Gorosabel et. al 2004 Redshift distribution Bromm & Loeb 2002 Redshift distribution

74 Mörtsell & Sollerman 2005!Without very low-z training set, good coverage at low-z, GRBs alone sensitive to Ω M, not EOS parameter (w = w o ) or time variation! Unlikely that enough low z events will be found even in Swift era Gorosabel et. al 2004 Redshift distribution Bromm & Loeb 2002 Redshift distribution (plots assume flatness prior)

75 Xu, Dai, & Liang 2005 Method I Method II Method III Dashed 17 GRBs (Method II) Blue 156 SNe Ia (Gold) Colored Combined 17 GRBs Method I: η, κ re-fit for each cosmology Method II: η, κ marginalized over Method III: η, κ fixed (theory or future low-z training set)