A Cosmology Independent Calibration of GRB Luminosity Relations and the Hubble Diagram

1. A Cosmology Independent Calibration of GRB Luminosity Relations and the Hubble Diagram Speaker: Liang Nan Collaborators: Wei Ke Xiao, Yuan Liu, Shuang Nan Zhang Tsinghua Center for Astrophysics , Tsinghua University National Cosmology Workshop “Constraints on Dark Energy by including GRBs” Dec, 10th, 2008, IHEP

2. Outline • (Ⅰ) Introduction We present a new method to calibrate GRB relations in a cosmology independent way. • (Ⅱ) Calibration of GRB Luminosity Relations We obtain the distance modulus of a GRB at a given redshift from the SNe Ia data and calibrate five GRB relations without assuming a particular cosmological model. • (Ⅲ) The Hubble Diagram of GRBs We construct the GRB Hubble diagram to constrain cosmological parameters. • (Ⅳ) Summary and Discussion

3. (Ⅰ) Introduction • Gamma-Ray Bursts (GRBs) are the most intense explosions observed so far. → GRBs are likely to occur in high redshift range. • GRB luminosity relations are connections between measurable properties of the γ-ray emission with the luminosity or energy. →Recent years, several power law GRB relations have been proposed in many works. →Many authors have made use of GRB relations as “standard candles” at very high redshift for cosmology research. see e.g. Ghirlanda, Ghisellini, & Firmani (2006),Schaefer (2007) for reviews

4. Cosmology-dependent Calibration of GRB relations • The most important point related to the use of GRBs in cosmology: ---- The dependence of the cosmological model of the calibration of GRB relations. • SN Ia cosmology: ---- adequate sample at low-redshift which can be used to calibrate the Phillips relation essentially independent of any cosmology. • GRB cosmology : ---- difficult to calibrate the relations using a low-z sample. ---- Calibration of GRB relations so far have been derived by assuming a particular cosmology (e.g. the ΛCDM model).

5. The circularity problem • In order to investigate cosmology, the relations of standard candles should be calibrated in a cosmological model independent way. ----Otherwise the circularity problem can not be avoided easily. • In principle, the circularity problem can be avoided in two ways (Ghirlanda et al. 2006): (i) A solid physical interpretation of these relations which would fix their slope independently from cosmology. (ii) The calibration of these relations by several low redshift GRBs.

6. Statistical approaches • Many previous works treated the circularity problem by means of statistical approaches. →simultaneous fit (Schaefer 2003:) the parameters in the calibration curves and the cosmology should be carried out at the same time. → Bayesian method (Firmani et al. 2005: ) → Markov Chain Monte Carlo global fitting (Li et al. 2008 ) ---- Because a particular cosmology model is required in doing the joint fitting, the circularity problem is not solved completely by means of statistical approaches.

7. Cosmic distance ladder • SN Ia cosmology: the distance of nearby SN Ia used to calibrate the Phillips relation can be obtained by measuring Cepheids in the same galaxy. Thus Cepheids has been regard as the first order standard candle to calibrate SNe Ia as the secondly order standard candle. • Distance of SN Ia obtained directly from observations are completely cosmological model independent. • It is obvious that the sources at the same redshift should have the same luminosity distance for any certain cosmology. • If distance modulus of GRBs can be obtained direct from the SN Ia data, we can calibrate the relations of GRBs in a cosmology independent way. Cepheids → SNe Ia SNe Ia → GRBs

8. Cosmic distance ladder: Cepheids →SNe Ia → GRBs 1929: Discovery of cosmic expansion Extra- galactic Cepheids Low-z GRB Standard Candle High-z GRB Cosmology Galactic Cepheids   Nearby SN Ia Standard Candle Low-z SN Ia Cosmology  1998: Discovery of Dark Energy <0.1 ~1.7 6.6 z

9. Methods • There are so many SN Ia samples that we can obtain the distance modulus at any redshift in the redshift range of SN Ia directly from the Hubble diagram of SN Ia. →InterpolationMethod → Iterative Method On the basis of smoothing the noise of supernova data over redshift, a non-parametric method in a model independent manner to reconstruct the luminosity distance at any redshift in the redshift range of SNe Ia.

10. If regarding the SN Ia as the first order standard candle, we can obtain the distance modulus of GRBs in the redshift range of SN Ia andcalibrate the relations of GRBs in a completely cosmology independent way. • By utilizing the relations to the GRB data at high redshift, we can use the standard Hubble diagram method to constrain the cosmological parameters.

11. (Ⅱ) Calibration of GRB Luminosity Relations • We adopt the data of 192 SNe Ia (Davis et al. 2007). • SN1997ff (z = 1.755): the only one SN Ia point atz>1.4, therefore we initially exclude it from our SN Ia sample. • With the GRB sample at z<1.4obtained from SNe Ia data, GRB luminosity relations can be calibrated in a completely cosmology independent way.

12. Methods → InterpolationMethod (the GRB sample at z<1.4:interpolating from SNe Ia data) → Iterative Method (the GRB sample: iterating from SNe Ia data) We follow the iterative procedure and adopt the results from Wu & Yu, 2008 The best fitting result is obtained by minimizing

13. GRB Luminosity Relations Calibration A GRB luminosity relation can be generally written in the form of Five GRB Relations (Schaefer 2007, 69 GRBs)

14. (1)lag - L relation(Norris, Marani, & Bonnell 2000) (Schaefer 2007, 38 GRBs)

15. (2) Variability - L relation (Fenimore & Ramirez-Ruiz 2000) (Schaefer 2007, 51 GRBs)

16. (3) Epeak- L relation (Schaefer 2003; Yonetoku et al. 2004) (Schaefer 2007, 64 GRBs)

17. (4) Eγ -Epeak relation (Ghirlanda, Ghisellini & Lazzati 2004) the so-called Ghirlanda relation (Schaefer 2007, 27 GRBs)

18. (5) τRT - L relation (Schaefer, 2007) τRT : the minimum rise time in the GRB light curve (Schaefer 2007, 62 GRBs)

19. Table 1. Calibration results (a: intercept, b: slope) for the seven GRB relations with the sample at z < 1.4 by using two methods from SNe Ia data and by assuming the LCDM. → Bisector: In order to avoid specifying “dependent” and “independent” variables → OLS (ordinary least-squares) → WLS (weighted least-squares) : taking into account the measurable uncertainties, results in almost identical best fits.

20. (Ⅲ) The Hubble Diagram of GRBs • We use the same method used in Schaefer (2007) to obtain the best estimate μ for each GRB which is the weighted average of all available distance moduli. • The derived distance modulus for each GRB is with its uncertainty where the summations run from 1 to 5 over the five relations used in Schaefer (2007) with available data.

21. Fig 1.Hubble Diagram of SNe Ia and GRBs Concordance model z = 1.4 SN1997ff(z = 1.755) → GRB high-z “data” , obtained from the calibrated GRB “standard” candles; These data are used to fit cosmological parameters at high-z. → GRB low-z “data”, obtained from SN Ia data, (thus also cosmology independent). These data are used to calibrate the GRB “standard” candles. → SNe Ia data(Davis et al. 2007), directly from observations, cosmology independent. These data used to obain the distance moduli of GRB low-z “data”, blue stars：Iterative Method Black circles：InterpolationMethod

23. (a)：InterpolationMethod (b)：Iterative Method Cosmological Constrains : ΛCDM model Chisquare statistic for the flat ΛCDM model → WLS Biesctor

24. Cosmological Constrains : WCDM model For the dark energy model with a constant equation of state (a)：InterpolationMethod (b)：Iterative Method → WLS Biesctor

25. (Ⅳ) Summary and Discussion • With the basic assumption that objects at the same redshift should have the same luminosity distance, we can obtain the distance modulus of a GRB at given redshift by interpolating from the Hubble diagram of SNe Ia at z <1.4. • There are not significant differences between the calibration results obtained by using our methods and those calibrated by assuming one particular cosmological model with the same GRB sample at z <1.4. This is not surprising, since the cosmological model is consistent with the data of SNe Ia. • We stress again that the luminosity relations we obtained here are cosmology independent completely.

26. We construct the GRB Hubble diagram and constrain cosmological parameters by the minimum χ2 method as in SN Ia cosmology. →for the flat ΛCDM model InterpolationMethodΩM = 0.28+0.05−0.06 IterativeMethodΩM = 0.31+0.06−0.06 →for DE model with a constant EoS InterpolationMethodw0 = −0.94+0.27−0.45 IterativeMethodw0 = −0.82+0.24−0.36 ----Our result suggests the the concordance model (w0 = −1,Ωm = 0.27, ΩΛ = 0.73) which is mainly derived from observations of SNe Ia at lower redshift is still consistent with the GRB data at higher redshift up to z = 6.6.

27. Further examinations should be required for considering GRBs as standard candles to cosmological use. → It is possible that some unknown biases of SN Ia luminosity relations may propagate into the interpolation results. Nevertheless, the observable distance of SNe Ia used in our methods to calibrate the relations of GRBs are completely independent of any given cosmological model. → Recently, there are possible evolution effects and selection bias claimed in GRB relations. • We emphasize that our method avoids the circularity problem more clearly than previous cosmology dependent calibration methods. Therefore our results on these GRB relations are less dependent on a prior cosmology.