Gamma-Ray Bursts are precise distance indicators similar to Type Ia Supernovae?

Gamma-Ray Bursts are precise distance indicators similar to
Type Ia Supernovae?

[    [    [    [    [
Abstract

We estimate the distance modulus to long gamma-ray bursts (LGRBs) using the Type I Fundamental Plane, a correlation between the spectral peak energy , the peak luminosity , and the luminosity time ( where is isotropic energy) for small Absolute Deviation from Constant Luminosity(ADCL). The Type I Fundamental Plane of LGRBs is calibrated using 8 LGRBs with redshift . To avoid any assumption on the cosmological model, we use the distance modulus of 557 Type Ia supernovae (SNeIa) from the Union 2 sample. This calibrated Type I Fundamental Plane is used to measure the distance moduli to 9 high-redshift LGRBs with the mean error , which is comparable with that of SNe Ia where stands for the distance modulus. The Type I Fundamental Plane is so tight that our distance moduli have very small uncertainties. From those distance moduli, we obtained the constraint for flat CDM universe. Adding 9 LGRBs distance moduli () to 557 SNeIa distance moduli () significantly improves the constraint for non-flat CDM universe from ()=(, ) for SNeIa only to ()=(, ) for SNeIa and 9 LGRBs.

a]Ryo Tsutsui, b]Takashi Nakamura, c]Daisuke Yonetoku, d]Keitaro Takahashi, c]and Yoshiyuki Morihara

Prepared for submission to JCAP

Gamma-Ray Bursts are precise distance indicators similar to

Type Ia Supernovae?


  • Research Center for the Early Universe, School of Science, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan

  • Department of Physics, Kyoto University, Kyoto 606-8502, Japan

  • Department of Physics, Kanazawa University, Kakuma, Kanazawa, Ishikawa 920-1192, Japan

  • Faculty of Science, Kumamoto University, Kurokami, Kumamoto, 860-8555, Japan

E-mail: tsutsui@resceu.s.u-tokyo.ac.jp, takashi@tap.scphys.kyoto-u.ac.jp, yonetoku@astro.s.kanazawa-u.ac.jp, keitaro@sci.kumamoto-u.ac.jp, morihara@astro.s.kanazawa-u.ac.jp

Keywords: gamma rays: bursts — gamma rays: observations— gamma rays: cosmology

ArXiv ePrint: 1205.2954

 

 

1 Introduction

Since the discoveries of possible distance indicators of long gamma-ray bursts (LGRBs) [1, 2, 3, 4, 5, 6, 7], many attempts to measure the distance up to the high redshift universe () which cannot be probed by Type Ia supernovae have been done to test cosmological models [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. These attempts are very important since no other distance indicators have been known in such high redshift universe.

However the large statistical and systematic errors of the distance indicators restricted the precision of distance measurements by LGRBs. To estimate distances of LGRBs more precisely, there are two directions : to discover more precise distance indicator [10, 21, 6] and/or to take weighted average of distances derived by many distance indicators [11, 13, 16]. These studies, however, are on the verge of many difficulties, such as missing or multiple jet breaks in Swift era [22, 23] or increasing scatters of distance indicators [24, 25, 26]. To make LGRBs to be more precise and reliable standard candles, we must know the origins of the dispersion of distance indicators to control them carefully.

In our previous studies, we have investigated origins of dispersion of the spectral - brightness correlations and found possible origins of systematic errors [27, 28]: (i) spectral peak energies () estimated by fitting the spectra with cut-off power-law model with three free parameters tend to be overestimated compared with those obtained by using the Band model with four free parameters [29], (ii) peak luminosities () should be estimated for a fixed time scale in the individual rest frame rather than the observer frame, (iii) contamination of short GRBs or other unknown populations. Removing the systematic errors by (i) and (iii), and correcting (ii), we found that the Fundamental Plane of LGRBs, that is, a correlation between the spectral peak energy , the rest frame 2.752-second peak luminosity , and the luminosity time ( where is the isotropic energy), is much tighter than the original correlation by R. Tsutsui et al. [17] [See also 30].

Recently, we found that dividing LGRBs into two groups according to the absolute deviation from their constant luminosity () significantly improves the Fundamental Plane of LGRBs [31]. is of critical importance in this work so let us summarize the main results below. is a quantity which characterizes the overall shape of light curve of a GRB and defined as follows:

(1.1)

where is normalized cumulative counts and is the normalized time by in which the cumulative counts increase from 0.01 to 0.99. Most of small- events consist of many pulses with almost the same brightness, while most of large- events consist of a bright main pulse and the weak long tail like FRED (Fast Rise and Exponential Decay). In Figure. 1, we show the cumulative light curves of LGRBs from Tsutsui, et al. [31]. The time and the cumulative counts are normalized by and the total counts, respectively. The red solid lines indicate small- events while blue solid lines do large- events. The black dashed line indicates the line of the virtual source with a constant luminosity. We clearly see that red lines with small ADCL () and blue solid lines with large ADCL() form different groups in this figure so that our classification scheme for subclasses of LGRB would be reasonable.

In Tsutsui, et al. [31], we found that the best-fit function for small ADCL() events is given by

(1.2)

with and , while that for large ADCL() events is given by

(1.3)

with and . Since both best-fit functions separately form the plane in space, we call the best fit function for small- (large-) as Type I (Type II) Fundamental Plane, respectively. Figure. 2 shows the Type I (left) and Type II (right) Fundamental Planes for given cosmological parameters from Tsutsui, et al. [31]. In both figures, Small- events are marked with red filled circles and large- events are with blue squares. Orange and light blue squares indicate outliers from the Type I and Type II Fundamental Planes, respectively. (see [31] for the definition of outliers. ) The left (right) of Figure. 2 is the plane perpendicular to the Type I (Type II) Fundamental Plane so that the Type I (Type II) Fundamental Plane is expressed by the solid line in each figure. In the left of Figure. 2, blue triangles which belong to the large ADCL events are above the solid line (=the Type I Fundamental Plane) while in the right of Figure. 2, red filled circles which belong to the small ADCL events are below the solid line (=the Type II Fundamental Plane) so that the existence of two Fundamental Planes can be recognized. The existence of such two subclasses is very similar to that of two Period - Luminosity relations of Cepheid variables [32].

Figure 1: The cumulative light curves of LGRBs from Tsutsui, et al. [31]. The red solid lines indicate small- events while blue solid lines do large- events. The black dashed line indicates the line of the virtual source with the constant luminosity. We clearly see that red lines with small ADCL () and blue solid lines with large ADCL() form different groups.
Figure 2: The diagram both for the Type I (left) and Type II (right) Fundamental Planes for given cosmological parameters from Tsutsui, et al. [31]. Small- events are marked with red filled circles and large- events with blue squares. Except for outliers ( orange triangles), most of small- events are on the Type I Fundamental Plane (=the solid line in the left figure) within their error bars without any systematic errors, while most of large- events are on the Type II Fundamental Plane (=the solid line in the right figure) except for outliers (light blue triangles). (see [31] for the definition of outliers. ) In the left figure, blue squares which belong to large ADCL events are above the solid line (=the Type I Fundamental Plane) while in the right figure, red filled circles which belong to small ADCL events are below the solid line (=the Type II Fundamental Plane) so that the existence of two Fundamental Planes can be recognized. The existence of such two subclasses is very similar to that of two Period - Luminosity relations of Cepheid variables.

In this paper using 21 small- LGRBs from the 36 gold sample of Tsutsui et al. [31], we try to measure distance modulus up to and determine the cosmological parameters. To avoid any assumption on cosmological models, we use the distance moduli of 557 Type Ia supernovae (SNeIa) from the Union 2 sample to calibrate the Type I Fundamental Plane of LGRBs by the 10 low-z events in . This is a big difference from our previous paper [31] where cosmological parameters were assumed to derive the relation. Applying the Type I Fundamental Plane calibrated at low redshifts without cosmological parameters to high-z GRBs, we obtain the relation between the redshift and the distance modulus of 11 high-z events in . From these distance moduli, we will obtain the constraint on for flat CDM universe. In this paper we do not use the Type II Fundamental Plane as a distance indicator to constrain the cosmological parameters due to the lack of enough number of large ADCL events. Throughout the paper, we fix the Hubble parameter as km s Mpc .

2 Calibration

2.1 local regression

First, we calibrate the Type I Fundamental Plane of LGRBs using low-z events whose distance moduli can be obtained by those of Type Ia supernovae. Actually, to estimate the distance moduli of low-z GRBs, we use a local regression method first developed by W. S. Cleveland. [33] and W. S. Cleveland and S. J. Devlin. [34], and first applied for GRB cosmology by V. F. Cardone et al. [16]. This method does not need a global shape function of redshift-distance modulus relation and is one of interpolation procedures which use only the data near points and fit them with a low-degree polynomial. Although the number of data and the degree of polynomial are arbitrary variables, it does not depend on a shape of a global function.

Applying this method, we obtain the distance moduli of 10 low-redshift LGRBs with mean error where stands for the distance modulus.

2.2 calibration

Having estimated the distance moduli of 10 low-redshift GRBs in a model independent way, we can calibrate the Type I Fundamental Plane of GRBs. Let us assume a linear correlation between , and in logarithmic scale as,

(2.1)

where , and are the parameters of the model and and are logarithmic mean of and with keV and sec, respectively. From equation (2.1), we obtain given by,

(2.2)

where is the observed energy flux. We define a chi square function,

(2.3)

and

(2.4)

where is the systematic error determined by iteration.

Before we minimize the chi square function, we use the robust statistic and outlier detection method which have been developed in Tsutsui et al. [30]. As a result, we detected two outliers (091003, 080319B) , and removed them from following chi square analysis. Performing chi square regression on the remaining samples, we obtained the best fit parameters and 1- uncertainties summarized in table. 1. The parameters in table. 1 and outliers of the relation are consistent with those in Tsutsui et al. [31]. Figure. 3 shows the diagram for small- events in . Two outliers are marked with green triangles and the solid line indicates the best fit function of Eq.(2.1) with values of A, B and C in table 1.

Figure 3: The diagram for small- events (red filled circles) in without any assumption of cosmological parameters. The solid line is the best fit function with and . The green triangles are outliers. (see [31] for the definition of outliers.)
A B C
52.06(0.02) 1.90(0.05) 0.25(0.13) 0 0.83
Table 1: The best fit values (1- uncertainties) of the model parameters of the Type I Fundamental Plane (Eq. (2.1)).
GRB
030329 0.168 39.48 0.04 39.61 0.17 0.13
090618 0.54 42.43 0.04 42.33 0.19 0.10
050525 0.606 42.83 0.07 42.72 0.13 0.13
970228 0.695 43.15 0.05 42.7 0.72 0.14
990705 0.843 43.75 0.06 43.86 0.22 0.10
091208B 1.063 44.3 0.09 44.77 0.77 0.07
061007 1.261 44.73 0.06 44.78 0.16 0.14
990506 1.3 44.81 0.07 44.82 0.26 0.08
070125 1.547 - - 45.39 0.25 0.10
990123 1.6 - - 45.5 0.16 0.09
990510 1.619 - - 45.46 0.34 0.14
090902B 1.822 - - 45.91 0.15 0.09
081121 2.512 - - 46.9 0.24 0.07
080721 2.602 - - 46.6 0.28 0.16
050401 2.9 - - 46.5 0.42 0.15
971214 3.418 - - 47.5 0.73 0.12
090323 3.57 - - 48.05 0.3 0.12
091003 0.897 43.82 0.09 45.97 0.28 0.16
080319B 0.937 43.82 0.08 45.36 0.17 0.14
110213A 1.46 - - 42.9 0.38 0.09
110422A 1.77 - - 43.51 0.16 0.10
Table 2: The redshifts, distance modulus of LGRBs estimated by local regression method and the ones estimated from the Type I Fundamental Plane. The first through the seventh column express name of the GRB, the redshift, the distance modulus determined by TypeIa supernovae, the error of , the distance modulus determined by the best fit function of Type I Fundamental Plane (Eq. (2.1)), the error of , the value of , respectively. Four outliers are separated in the last four rows. They are the same as the outliers for small- GRBs in Tsutsui et al. [31]

3 Constraint

Applying equation (2.2) to high redshift LGRBs, we can obtain the extended Hubble diagram with the unprecedented precision. We define as below,

Here we neglect the effect of the gravitational lensing because M. Oguri and K. Takahashi. [35] discussed the gravitational lensing of GRBs and found that the biases are small.

Then we derive constraints on cosmological parameters. In the -CDM model, the luminosity distance () is given as a function of the density parameters, and .

The chi square function is defined by,

(3.2)

where . We used the robust regression and outlier detection method again. As a result, we detect two outliers (110213A, 110422A), and removed it from following chi square analysis. In table 2, we summarize the redshifts, distance modulus of LGRBs estimated by local regression method and the ones estimated from the Type I Fundamental Plane with their 1- uncertainties.

Figure 4: Extended Hubble diagram from the Type I Fundamental Plane. Gray squares: the 557 SNe Ia from R. Amanullah et al.[36]. Blue open circlues: LGRBs at estimated from the local regression of SNe Ia data. red filled circles: LGRBs estimated from the Type I Fundamental Plane. Green triangles: outliers. The solid line indicates a theoretical model calculated with ()=(0.30, 0.70). Inset : A residual Hubble diagram from the theoretical model.

Figure 4 shows an extended Hubble diagram up to . The gray squares indicate the SNe Ia from R. Amanullah et al. [36], the blue open circles indicate the low redshift LGRBs at estimated from the local regression of SNe Ia data. Red filled circles and green triangles indicate LGRBs estimated from the Type I Fundamental Plane, and outliers, respectively. The solid line indicates a theoretical model calculated with ()=(0.30,0.70). As Figure 4 shows, the uncertainties of distance modulus of LGRBs are as small as those of individual SNe Ia (gray squares). The mean error of distance modulus of LGRBs are , while those of SNe Ia are .

Performing the chi square analysis on the remaining 9 LGRBs, we obtained the constraint for flat CDM universe. In Figure 5, we show the likelihood contour for non-flat CDM universe from 9 LGRBs (green), with the likelihood contours from 557 SNe Ia (blue) and 9 LGRBs + 557 SNe Ia (red), and the best-fit values with 1- errors are shown in table 3. Figure 4 shows that adding only 9 high redshift LGRBs () to 557 SNe Ia () significantly improves the constraint on plane.

Figure 5: Constraints on () plane from 9 LGRBs with (green), 557 SNe Ia (blue) and 9 LGRBs + 557 SNe Ia (red). The contours correspond to 68.3%, 95.4%, 99.7% confidence regions. See also Table 3.
/d.o.f
9 LGRBs (flat) - 5.09/7

557 SNe Ia (non-flat)
542.1/555
Combined (non-flat) 548.3/564
Table 3: Constraints on () for flat universe from 9 LGRBs and for non-flat universe from 557 SNe Ia and 9 LGRBs + 557 SNe Ia with their reduced chi squares. All the quoted errors are at the 1- confidence level.

4 Discussion

In this paper we extended the Hubble diagram with the Type I Fundamental Plane of LGRBs. Because of the tightness of the relation, our distance measurements are much more precise than those of previous works. Although the number of LGRBs are much smaller than that of the previous works [11, 12, 13, 14, 15, 16, 17], our constraint is much stronger. This indicates the importance of controlling and removing systematic errors. Although we used only the Type I Fundamental Plane to measure distance modulus, in principle, we may also use the Type II Fundamental Plane as an independent distance indicator. However, because there is still a small number of long tailed events, it is difficult to calibrate the relation with our outlier rejection technique.

The constraints from 9 high redshift GRBs are consistent with those from other probes, e.g. Cosmic Microwave Background (CMB), Baryon Acoustic Oscillation (BAO)[37, 38]. We should stress here that this consistency is not trivial. Since the redshift range of GRB () is different from TypeIa supernovae (), BAO () and CMB (), they might show different cosmological parameters if the dark energy is strongly time dependent. Therefore our results suggest that time dependence of the dark energy is not so strong even if it exists.

The purpose of this paper is to show the potential high ability of Type I Fundamental Plane of the LGRBs as a distance indicator. Combining our data with other cosmological probes is the next step. A different direction is to test more general equation of state than CDM universe. We are now trying to combine our data with CMB and BAO as well as more general equation of state to constrain the property of the dark energy.

Acknowledgments

This work is supported in part by the Grant-in-Aid for Grant-in-Aid for Young Scientists (B) from the Japan Society for Promotion of Science (JSPS), No.24740116(RT), by the Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, No.23540305 (TN), No.20674002 (DY), No.23740179 (KT), and by the Grant-in-Aid for the global COE program The Next Generation of Physics, Spun from Universality and Emergence at Kyoto University.

References

  • [1] E. E. Fenimore and E. Ramirez-Ruiz, Redshifts for 220 batse gamma-ray bursts determined by variability and the cosmological consequences, eprint arXiv (Apr, 2000) 4176.
  • [2] J. P. Norris, G. F. Marani, and J. T. Bonnell, Connection between energy-dependent lags and peak luminosity in gamma-ray bursts, The Astrophysical Journal 534 (May, 2000) 248.
  • [3] L. Amati, F. Frontera, M. Tavani, J. J. M. in’t Zand, A. Antonelli, E. Costa, M. Feroci, C. Guidorzi, J. Heise, N. Masetti, E. Montanari, L. Nicastro, E. Palazzi, E. Pian, L. Piro, and P. Soffitta, Intrinsic spectra and energetics of bepposax gamma-ray bursts with known redshifts, Astronomy and Astrophysics 390 (Jul, 2002) 81.
  • [4] D. Yonetoku, T. Murakami, T. Nakamura, R. Yamazaki, A. K. Inoue, and K. Ioka, Gamma-ray burst formation rate inferred from the spectral peak energy-peak luminosity relation, The Astrophysical Journal 609 (Jul, 2004) 935.
  • [5] G. Ghirlanda, G. Ghisellini, and D. Lazzati, The collimation-corrected gamma-ray burst energies correlate with the peak energy of their spectrum, The Astrophysical Journal 616 (Nov, 2004) 331.
  • [6] E. Liang and B. Zhang, Model-independent multivariable gamma-ray burst luminosity indicator and its possible cosmological implications, The Astrophysical Journal 633 (Jan, 2005) 611–611–623–623.
  • [7] C. Firmani, G. Ghisellini, V. Avila-Reese, and G. Ghirlanda, Discovery of a tight correlation among the prompt emission properties of long gamma-ray bursts, Monthly Notices of the Royal Astronomical Society 370 (Jul, 2006) 185.
  • [8] J. S. Bloom, D. A. Frail, and S. R. Kulkarni, Gamma-ray burst energetics and the gamma-ray burst hubble diagram: Promises and limitations, The Astrophysical Journal 594 (Sep, 2003) 674.
  • [9] B. E. Schaefer, Gamma-ray burst hubble diagram to z=4.5, The Astrophysical Journal 583 (Feb, 2003) L67.
  • [10] G. Ghirlanda, G. Ghisellini, D. Lazzati, and C. Firmani, Gamma-ray bursts: New rulers to measure the universe, The Astrophysical Journal 613 (Sep, 2004) L13.
  • [11] B. E. Schaefer, The hubble diagram to redshift from 69 gamma-ray bursts, The Astrophysical Journal 660 (May, 2007) 16.
  • [12] Y. Kodama, D. Yonetoku, T. Murakami, S. Tanabe, R. Tsutsui, and T. Nakamura, Gamma-ray bursts in suggest that the time variation of the dark energy is small, Monthly Notices of the Royal Astronomical Society: Letters 391 (Nov, 2008) L1.
  • [13] N. Liang, W. K. Xiao, Y. Liu, and S. N. Zhang, A cosmology-independent calibration of gamma-ray burst luminosity relations and the hubble diagram, The Astrophysical Journal 685 (Sep, 2008) 354.
  • [14] L. Amati, C. Guidorzi, F. Frontera, M. D. Valle, F. Finelli, R. Landi, and E. Montanari, Measuring the cosmological parameters with the correlation of gamma-ray bursts, Monthly Notices of the Royal Astronomical Society 391 (Dec, 2008) 577.
  • [15] R. Tsutsui, T. Nakamura, D. Yonetoku, T. Murakami, S. Tanabe, Y. Kodama, and K. Takahashi, Constraints on w0 and wa of dark energy from high-redshift gamma-ray bursts, Monthly Notices of the Royal Astronomical Society: Letters 394 (Mar, 2009) L31.
  • [16] V. F. Cardone, S. Capozziello, and M. G. Dainotti, An updated gamma-ray bursts hubble diagram, Monthly Notices of the Royal Astronomical Society 400 (Dec, 2009) 775.
  • [17] R. Tsutsui, T. Nakamura, D. Yonetoku, T. Murakami, Y. Kodama, and K. Takahashi, Cosmological constraints from calibrated yonetoku and amati relation suggest fundamental plane of gamma-ray bursts, Journal of Cosmology and Astroparticle Physics 08 (Aug, 2009) 015.
  • [18] H. Wei and S.-N. Zhang, Reconstructing the cosmic expansion history up to redshift z=6.29 with the calibrated gamma-ray bursts, The European Physical Journal C 63 (Sep, 2009) 139.
  • [19] H. Wei, Observational constraints on cosmological models with the updated long gamma-ray bursts, Journal of Cosmology and Astroparticle Physics 08 (Aug, 2010) 020.
  • [20] A. Diaferio, L. Ostorero, and V. Cardone, Gamma-ray bursts as cosmological probes: Λcdm vs. conformal gravity, Journal of Cosmology and Astroparticle Physics 10 (Oct, 2011) 008.
  • [21] C. Firmani, V. Avila-Reese, G. Ghisellini, and G. Ghirlanda, The hubble diagram extended to : the gamma-ray properties of gamma-ray bursts confirm the Λ cold dark matter model, Monthly Notices of the Royal Astronomical Society: Letters 372 (Oct, 2006) L28.
  • [22] G. Ghirlanda, L. Nava, G. Ghisellini, and C. Firmani, Confirming the -ray burst spectral-energy correlations in the era of multiple time breaks, Astronomy and Astrophysics 466 (Apr, 2007) 127.
  • [23] G. Sato, R. Yamazaki, K. Ioka, T. Sakamoto, T. Takahashi, K. Nakazawa, T. Nakamura, K. Toma, D. Hullinger, M. Tashiro, A. M. Parsons, H. A. Krimm, S. D. Barthelmy, N. Gehrels, D. N. Burrows, P. T. O’Brien, J. P. Osborne, G. Chincarini, and D. Q. Lamb, Swift discovery of gamma-ray bursts without a jet break feature in their x-ray afterglows, The Astrophysical Journal 657 (Mar, 2007) 359.
  • [24] N. R. Butler, D. Kocevski, J. S. Bloom, and J. L. Curtis, A complete catalog of swift gamma-ray burst spectra and durations: Demise of a physical origin for pre-swift high-energy correlations, The Astrophysical Journal 671 (Dec, 2007) 656.
  • [25] F. Rossi, C. Guidorzi, L. Amati, F. Frontera, P. Romano, S. Campana, G. Chincarini, E. Montanari, A. Moretti, and G. Tagliaferri, Testing the -- correlation on a BeppoSax and Swift sample of gamma-ray bursts, Monthly Notices of the Royal Astronomical Society 388 (Aug, 2008) 1284.
  • [26] A. C. Collazzi and B. E. Schaefer, Does the addition of a duration improve the liso-epeak relation for gamma-ray bursts?, The Astrophysical Journal 688 (Nov, 2008) 456.
  • [27] D. Yonetoku, T. Murakami, R. Tsutsui, T. Nakamura, Y. Morihara, and K. Takahashi, Possible origins of dispersion of the peak energy-brightness correlations of gamma-ray bursts, Publications of the Astronomical Society of Japan 62 (Dec, 2010) 1495. (c) 2010: Astronomical Society of Japan.
  • [28] R. Tsutsui, T. Nakamura, D. Yonetoku, T. Murakami, and K. Takahashi, Intrisic dispersion of correlations among , , and of gamma ray bursts depends on the quality of data set, eprint arXiv:1012.3009 (Dec, 2010). 11 pages, 1 figure, submitted to ApJ Letter.
  • [29] D. Band, J. Matteson, L. Ford, B. Schaefer, D. Palmer, B. Teegarden, T. Cline, M. Briggs, W. Paciesas, G. Pendleton, G. Fishman, C. Kouveliotou, C. Meegan, R. Wilson, and P. Lestrade, Batse observations of gamma-ray burst spectra. i - spectral diversity, Astrophysical Journal 413 (Aug, 1993) 281.
  • [30] R. Tsutsui, T. Nakamura, D. Yonetoku, T. Murakami, Y. Morihara, and K. Takahashi, Improved -- diagram and a robust regression method, Publications of the Astronomical Society of Japan 63 (Aug, 2011) 741. (c) 2011: Astronomical Society of Japan.
  • [31] R. Tsutsui, T. Nakamura, D. Yonetoku, K. Takahashi, and Y. Morihara, Identifying subclasses of long gamma-ray bursts with cumulative light curve morphology of prompt emissions, eprint arXiv 1201 (Jan, 2012) 2763. 10 pages, 7 figures, 2 tables. Submitted to PASJ.
  • [32] W. Baade, The period-luminosity relation of the cepheids, Publications of the Astronomical Society of the Pacific 68 (Feb, 1956) 5.
  • [33] W. S. Cleveland, Robust locally weighted regression and smoothing scatterplots, Journal of the American Statistical Association 74 (1979), no. 368 829–836.
  • [34] W. S. Cleveland and S. J. Devlin, Locally weighted regression: An approach to regression analysis by local fitting, Journal of the American Statistical Association 83 (1988), no. 403 596–610.
  • [35] M. Oguri and K. Takahashi, Gravitational lensing effects on the gamma-ray burst hubble diagram, Physical Review D 73 (Jun, 2006) 123002.
  • [36] R. Amanullah, C. Lidman, D. Rubin, G. Aldering, P. Astier, K. Barbary, M. S. Burns, A. Conley, K. S. Dawson, S. E. Deustua, M. Doi, S. Fabbro, L. Faccioli, H. K. Fakhouri, G. Folatelli, A. S. Fruchter, H. Furusawa, G. Garavini, G. Goldhaber, A. Goobar, D. E. Groom, I. Hook, D. A. Howell, N. Kashikawa, A. G. Kim, R. A. Knop, M. Kowalski, E. Linder, J. Meyers, T. Morokuma, S. Nobili, J. Nordin, P. E. Nugent, L. Östman, R. Pain, N. Panagia, S. Perlmutter, J. Raux, P. Ruiz-Lapuente, A. L. Spadafora, M. Strovink, N. Suzuki, L. Wang, W. M. Wood-Vasey, N. Yasuda, and T. S. C. Project, Spectra and hubble space telescope light curves of six type ia supernovae at and the union2 compilation, The Astrophysical Journal 716 (Jun, 2010) 712.
  • [37] E. Komatsu, K. M. Smith, J. Dunkley, C. L. Bennett, B. Gold, G. Hinshaw, N. Jarosik, D. Larson, M. R. Nolta, L. Page, D. N. Spergel, M. Halpern, R. S. Hill, A. Kogut, M. Limon, S. S. Meyer, N. Odegard, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright, Seven-year wilkinson microwave anisotropy probe (wmap) observations: Cosmological interpretation, The Astrophysical Journal Supplement 192 (Feb, 2011) 18.
  • [38] D. J. Eisenstein, I. Zehavi, D. W. Hogg, R. Scoccimarro, M. R. Blanton, R. C. Nichol, R. Scranton, H.-J. Seo, M. Tegmark, Z. Zheng, S. F. Anderson, J. Annis, N. Bahcall, J. Brinkmann, S. Burles, F. J. Castander, A. Connolly, I. Csabai, M. Doi, M. Fukugita, J. A. Frieman, K. Glazebrook, J. E. Gunn, J. S. Hendry, G. Hennessy, Z. Ivezić, S. Kent, G. R. Knapp, H. Lin, Y.-S. Loh, R. H. Lupton, B. Margon, T. A. McKay, A. Meiksin, J. A. Munn, A. Pope, M. W. Richmond, D. Schlegel, D. P. Schneider, K. Shimasaku, C. Stoughton, M. A. Strauss, M. SubbaRao, A. S. Szalay, I. Szapudi, D. L. Tucker, B. Yanny, and D. G. York, Detection of the baryon acoustic peak in the large-scale correlation function of sdss luminous red galaxies, The Astrophysical Journal 633 (Nov, 2005) 560.
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
248048
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description