Analysis of Fermi GRB T 90 distribution

Transcription

1 Analysis of Fermi GRB T 90 distribution Mariusz Tarnopolski Astronomical Observatory Jagiellonian University 23 July 2015 Cosmology School, Kielce Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

2 Presentation plan 1 Introduction and overview Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

3 Presentation plan 1 Introduction and overview 2 T 90 distributions of Fermi GRBs Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

4 Presentation plan 1 Introduction and overview 2 T 90 distributions of Fermi GRBs χ 2 tting Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

5 Presentation plan 1 Introduction and overview 2 T 90 distributions of Fermi GRBs χ 2 tting Maximum log-likelihood tting Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

6 Presentation plan 1 Introduction and overview 2 T 90 distributions of Fermi GRBs χ 2 tting Maximum log-likelihood tting 3 Hurst Exponents (HEs) & Machine Learning (ML) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

7 Presentation plan 1 Introduction and overview 2 T 90 distributions of Fermi GRBs χ 2 tting Maximum log-likelihood tting 3 Hurst Exponents (HEs) & Machine Learning (ML) 4 On the limit between short and long GRBs Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

8 Presentation plan 1 Introduction and overview 2 T 90 distributions of Fermi GRBs χ 2 tting Maximum log-likelihood tting 3 Hurst Exponents (HEs) & Machine Learning (ML) 4 On the limit between short and long GRBs 5 Conclusions Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

9 Satellites Introduction and overview CGRO/BATSE Swift/BAT RHESSI BeppoSAX/GRBM Fermi/GBM HETE-2/FREGATE INTEGRAL/SPI-ACS SUZAKU Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

10 Introduction and overview KONUS (Mazets et al. 1981) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

11 BATSE 1B Introduction and overview Kouveliotou et al. (1993) tted a quadratic function between the two peaks of 222 GRBs and determined its minimum to be at (1.2 ± 0.4) s, which rounded of to the next integer bin edge, is 2.0 s. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

12 Introduction and overview BATSE 3B (Horváth 1998) Counts logt 90 Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

13 Introduction and overview BATSE 4B (current) (Horváth 2002) 0.7 PDF MLE chi2 Horvath logt 90 Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

14 Introduction and overview Swift (Horváth et al. 2008) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

15 Introduction and overview Swift (Huja et al. 2009) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

16 Introduction and overview Swift (Huja & ípa 2009) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

17 Introduction and overview RHESSI ( ípa et al. 2009) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

18 Introduction and overview BeppoSAX (Horváth 2009) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

19 T90 distributions of Fermi GRBs χ 2 tting A mixture of Gaussians: f k = k A i N i (µ i, σ 2) i i=1 f k = k i=1 A 2πσi i exp ( (x µ i ) 2 2σ 2 i ) is tted to a histogram of log (T 90 ). A signicance level of α = 0.05 is adopted; 25 binnings are applied, dened by the bin widths w from 0.30 to 0.06 with a step of The corresponding number of bins range from 15 to 69. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

20 T90 distributions of Fermi GRBs χ 2 tting w A mixture of Gaussians: f k = k A i N i (µ i, σ 2) i i=1 f k = k i=1 A 2πσi i exp ( (x µ i ) 2 2σ 2 i is tted to a histogram of log (T 90 ). A signicance level of α = 0.05 is adopted; 25 binnings are applied, dened by the bin widths w from 0.30 to 0.06 with a step of The corresponding number of bins range from 15 to 69. ) w w w w Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41 logt 90

21 T90 distributions of Fermi GRBs χ 2 tting Table 1: Parameters of a two-gaussian t w i µ i σ i A i χ 2 p-val Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

22 T90 distributions of Fermi GRBs χ 2 tting 300 w w w w w w w w w w logt 90 Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41 logt 90

23 T90 distributions of Fermi GRBs χ 2 tting Table 2: Parameters of a three-gaussian t w i µ i σ i A i χ 2 p-val Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

24 T90 distributions of Fermi GRBs χ 2 tting χ 2 = χ 2 1 χ 2 2. = χ 2 ( ν) Table 3: Improvements of a three-gaussian over a two-gaussian t w χ 2 p-value Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

25 T90 distributions of Fermi GRBs χ 2 tting 2.5 BATSE 3B BATSE 4B Swift Swift Swfit RHESSI X Fermi logt Horváth 1998 Horváth 2002 Huja et al ípa et al Horváth 2009 Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

26 Results 1 T90 distributions of Fermi GRBs χ 2 tting T 90 distribution of Fermi GRBs is bimodal no evidence for a (phenomenological) third (intermediate) class Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

27 T90 distributions of Fermi GRBs Maximum log-likelihood tting It feels like a waste of data to bin 1600 events into a few dozens of bins. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

28 T90 distributions of Fermi GRBs Maximum log-likelihood tting It feels like a waste of data to bin 1600 events into a few dozens of bins. Having a distribution with a PDF given by f = f (x; θ) (possibly a mixture), where θ = {θ i } p is a set of parameters, the log-likelihood function is i=1 dened as L = N i=1 log f (x i ; θ), where {x i } N i=1 are the datapoints from the sample to which a distribution is tted. The tting is performed by searching a set of parameters θ for which the log-likelihood L is maximized. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

29 T90 distributions of Fermi GRBs Maximum log-likelihood tting For nested as well as non-nested models, the Akaike information criterion (AIC ) may be applied. The AIC is dened as AIC = 2p 2L p. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

30 T90 distributions of Fermi GRBs Maximum log-likelihood tting For nested as well as non-nested models, the Akaike information criterion (AIC ) may be applied. The AIC is dened as AIC = 2p 2L p. A preferred model is the one that minimizes AIC. The formulation of AIC penalizes the use of an overly excessive number of parameters, hence discourages overtting. Among candidate models with AIC i, let AIC min denote the smallest. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

31 T90 distributions of Fermi GRBs Maximum log-likelihood tting For nested as well as non-nested models, the Akaike information criterion (AIC ) may be applied. The AIC is dened as AIC = 2p 2L p. A preferred model is the one that minimizes AIC. The formulation of AIC penalizes the use of an overly excessive number of parameters, hence discourages overtting. Among candidate models with AIC i, let AIC min denote the smallest. Then, Pr i = exp ( i 2 ), where i = AIC min AIC i, can be interpreted as the relative (compared to AIC min ) probability that the i-th model minimizes the AIC. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

32 T90 distributions of Fermi GRBs Maximum log-likelihood tting A mixture of k standard normal (Gaussian) N (µ, σ 2 ) distributions: f (N ) k (x) = k i=1 A i ϕ ( x µ i σ i ) = k i=1 A i 2πσi exp ( (x µ i) 2 2σ 2 i ) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

33 T90 distributions of Fermi GRBs Maximum log-likelihood tting A mixture of k standard normal (Gaussian) N (µ, σ 2 ) distributions: f (N ) k (x) = k i=1 A i ϕ ( x µ i σ i ) = k i=1 A mixture of k skew normal (SN) distributions: f (SN) k (x) = k i=1 A i 2πσi exp ( (x µ i) 2 2A i ϕ ( x µ i x µ i ) Φ (α i ) σ i σ i 2σ 2 i ) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

34 T90 distributions of Fermi GRBs Maximum log-likelihood tting A mixture of k standard normal (Gaussian) N (µ, σ 2 ) distributions: f (N ) k (x) = k i=1 A i ϕ ( x µ i σ i ) = k i=1 A mixture of k skew normal (SN) distributions: f (SN) k (x) = k i=1 A i 2πσi exp ( (x µ i) 2 2A i ϕ ( x µ i x µ i ) Φ (α i ) σ i σ i A mixture of k sinh-arcsinh (SAS) distributions: f (SAS) (x) = k A i k σ [1 i + ( x µ i σ i ) ] βi cosh [β i sinh 1 ( x µ i σ i ) δ i ] i=1 F (SAS) (x) = k exp [ 1 k 2 sinh [β i sinh 1 ( x µ i σ i ) δ i ] 2 ] i=1 2σ 2 i ) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

35 T90 distributions of Fermi GRBs Maximum log-likelihood tting A mixture of k standard normal (Gaussian) N (µ, σ 2 ) distributions: f (N ) k (x) = k i=1 A i ϕ ( x µ i σ i ) = k i=1 A mixture of k skew normal (SN) distributions: f (SN) k (x) = k i=1 A i 2πσi exp ( (x µ i) 2 2A i ϕ ( x µ i x µ i ) Φ (α i ) σ i σ i A mixture of k sinh-arcsinh (SAS) distributions: f (SAS) (x) = k A i k σ [1 i + ( x µ i σ i ) ] βi cosh [β i sinh 1 ( x µ i σ i ) δ i ] i=1 F (SAS) (x) = k exp [ 1 k 2 sinh [β i sinh 1 ( x µ i σ i ) δ i ] 2 ] i=1 A mixture of k alpha-skew-normal (ASN) distributions: f (ASN) k (x) = k i=1 x µ (1 α i i A i 2 + α 2 i σ i ) σ 2 i ϕ ( x µ i σ i ) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41 )

36 T90 distributions of Fermi GRBs Maximum log-likelihood tting AIC number of components AIC vs. number of components in a mixture of standard normal distributions. The minimal value corresponds to a three-gaussian. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

37 T90 distributions of Fermi GRBs Maximum log-likelihood tting PDF a b c d e f g h Distributions tted to log T 90 data. Color dashed curves are the components of the (black solid) mixture distribution. The panels show a mixture of (a) two standard Gaussians, (b) three standard Gaussians, (c) two skew-normal, (d) three skew-normal, (e) two sinh-arcsinh, (f) three sinh-arcsinh, (g) one alpha-skew-normal, and (h) two alpha-skew-normal distributions. logt 90 Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

38 T90 distributions of Fermi GRBs Maximum log-likelihood tting 2 G 3 G 2 SN 3 SN 2 SAS 3 SAS 1 ASN 2 ASN AIC Pr AIC a b c d e f g h Pr AIC and relative probability (Pr ) for the models examined. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

39 Results 2 T90 distributions of Fermi GRBs Maximum log-likelihood tting Log-likelihood method supported the non-existence of a third (intermediate) component in the T 90 distribution of Fermi. A two-component mixture of skewed distributions (2-SN and 2-SAS) describes the data better than a three-gaussian. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

40 Hurst Exponents (HEs) & Machine Learning (ML) Methods HE denition HE is a measure of persistency/long-term memory/self-similarity of a process. Two ways of dening: 1 a process Y (t) (non-stationary) is self-similar with self-similarity parameter H, if Y (λt). = λ H Y (t) 2 a process X (t) (stationary) is self-similar if α (0, 2): lim ρ(t) τ α, τ α = 2 2H Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

41 Hurst Exponents (HEs) & Machine Learning (ML) Methods HE properties 0 < H 1 H = 0.5 for a random walk (Brownion motion) H < 0.5 for anti-persistent (anti-correlated, short memory) process H > 0.5 for persistent (correlated, long memory) process H = 1 for periodic time series fractal dimension D = 2 H Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

42 Hurst Exponents (HEs) & Machine Learning (ML) 8 6 Counts HE Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

43 Hurst Exponents (HEs) & Machine Learning (ML) Methods MVTS Light curves are binned in to narrow time bins. Optimum bin-width at which the non-statistical variability in the light curve becomes signicant. Prompt duration emission and equal duration of background region. Ratio of the variances of the GRB and the background divided by the bin-width as a function of bin-width. For binnings beyond the minimum the signal is indistinguishable from Poissonian uctuations. Left from minimum, signicant variability in the light curve may vanish (coarse binning). Optimum bin-width is at the minimum. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

44 Hurst Exponents (HEs) & Machine Learning (ML) Methods SVM Not probabilistic, but methods exist (probability calibration, e.g. distance to the hyperplane) to make it probabilistic. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

45 Hurst Exponents (HEs) & Machine Learning (ML) Methods Monte Carlo & SVM ( 22 ) = 231 subsamples from short GRBs; for each, 435 subsamples of (out of 46) long GRBs training set. Remaining validation set realisations. Success ratio r : r short and r long ; r tot = 2r short+4r long. 6 Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

46 Hurst Exponents (HEs) & Machine Learning (ML) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

47 Hurst Exponents (HEs) & Machine Learning (ML) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

48 Hurst Exponents (HEs) & Machine Learning (ML) Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

49 Hurst Exponents (HEs) & Machine Learning (ML) Results 3 1 H and MVTS alone give unsatisfactory classications 2 T 90 works as expected 3 (H, log MVTS) unsatisfactory 4 (H, log T 90 ) better than H and log T 90 alone 5 (log MVTS, log T 90 ) worse, better 6 complementing (log MVTS, log T 90 ) with HEs accuracy increased by 7%; comparable to T 90 alone. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

50 On the limit between short and long GRBs As in (Kouveliotou et al. 1993), the limitting T 90 value is found as a local minimum. ML t of a two-gaussian instead of a parabola. Datasets: BATSE 1B (for comparison; 226 events), BATSE current, Swift, BeppoSAX and Fermi ( events). Parameter errors: parametric bootstrap. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

51 On the limit between short and long GRBs PDF 0.5 (a) 0.6 (b) (c) (d) (e) logt 90 Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

52 On the limit between short and long GRBs Table 4: Parameters of the ts. Errors are estimated using the bootstrap method. Label Dataset N i µ i δµ i σ i δσ i A i δa i min. δmin. (a) BATSE 1B (b) BATSE current (c) Swift (d) BeppoSAX (e) Fermi Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

53 On the limit between short and long GRBs Results 4 Datasets from Swift and BeepoSAX are unimodal, hence no new limit may be inferred. BATSE 1B and Fermi are consistent with the conventional 2 s value. A limit of 3.38 ± 0.27 s was obtained for BATSE current. This leads to diminishing the fraction of long GRBs in the sampe by 4%. T 90 is an ambiguous GRB type indicator. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

54 Conclusions Conclusions Both χ 2 and ML tting lead to a bimodal T 90 distribution This is a hint that there are only two GRB classes Two types of two-component skewed distributions are a better t than a three-gaussian It is unlikely that the third, intermediate-duration, GRB class is a real physical phenomenon It was suggested (Zitouni 2015) that an assymetric T 90 distribution may be due to an assymetric distribution of envelope masses of the progenitors Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

55 Conclusions Conclusions Both χ 2 and ML tting lead to a bimodal T 90 distribution This is a hint that there are only two GRB classes Two types of two-component skewed distributions are a better t than a three-gaussian It is unlikely that the third, intermediate-duration, GRB class is a real physical phenomenon It was suggested (Zitouni 2015) that an assymetric T 90 distribution may be due to an assymetric distribution of envelope masses of the progenitors HE might serve as a GRB class indicator including it in the SVM scheme increased accuracy by 7% Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

56 Conclusions Conclusions Both χ 2 and ML tting lead to a bimodal T 90 distribution This is a hint that there are only two GRB classes Two types of two-component skewed distributions are a better t than a three-gaussian It is unlikely that the third, intermediate-duration, GRB class is a real physical phenomenon It was suggested (Zitouni 2015) that an assymetric T 90 distribution may be due to an assymetric distribution of envelope masses of the progenitors HE might serve as a GRB class indicator including it in the SVM scheme increased accuracy by 7% A division between short and long GRBs at T 90 of 3.38 s is more appropriate for the BATSE current dataset than the the conventional value of 2 s. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

57 References [1] Tarnopolski M., 2015, A&A, in press (arxiv: ) [2] Tarnopolski M., 2015 (arxiv: ) [3] Tarnopolski M., 2015 (arxiv: ) [4] Tarnopolski M., 2015 (arxiv: ) [5] Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41

58 References [1] Tarnopolski M., 2015, A&A, in press (arxiv: ) [2] Tarnopolski M., 2015 (arxiv: ) [3] Tarnopolski M., 2015 (arxiv: ) [4] Tarnopolski M., 2015 (arxiv: ) [5] Thank you for your attention. Mariusz Tarnopolski (AO JU) Fermi GRBs 23 July / 41