EDP Sciences
Free Access
Issue
A&A
Volume 585, January 2016
Article Number A68
Number of page(s) 9
Section Cosmology (including clusters of galaxies)
DOI https://doi.org/10.1051/0004-6361/201526485
Published online 18 December 2015

© ESO, 2015

1. Introduction

Gamma-ray bursts (GRBs) are the most violent explosions in the Universe, with the highest isotropic energy up to 1054 erg (for reviews, see Mészáros 2006; Zhang 2007; Gehrels et al. 2009). Thus, they can be detected to the edge of the visible Universe (Ciardi & Loeb 2000; Lamb & Reichart 2000; Wang et al. 2012). For instance, the spectroscopically confirmed redshift of GRB090423 is about 8.2 (Tanvir et al. 2009; Salvaterra et al. 2009). Therefore, they are promising probes for the high-redshift Universe (for a recent review, see Wang et al. 2015). Many studies have been carried out to use GRBs for cosmological purposes, such as the star formation rate (Totani 1997; Wijers et al. 1998; Porciani & Madau 2001; Wang & Dai 2009, 2011a), the intergalactic medium metal enrichment (Barkana & Loeb 2004; Wang et al. 2012), dark energy (Dai et al. 2004; Friedman & Bloom 2005; Schaefer 2007; Basilakos & Perivolaropoulos 2008; Wang et al. 2011b), reionization (Totani et al. 2006; Gallerani et al. 2008; Wang 2013), possible anisotropic acceleration (Wang & Wang 2014a), and the two-point correlation (Li & Lin 2015).

To constrain the cosmological parameters, standard rulers or candles such as baryon acoustic oscillations (BAO; Cole et al. 2005; Eisenstein et al. 2005; Anderson et al. 2014), cosmic microwave background (CMB; Komatsu et al. 2011; Planck Collaboration XVI 2013; Planck Collaboration XIII 2015) and SNe Ia (Riess et al. 1998; Perlmutter et al. 1999; Suzuki et al. 2012) are required. The redshifts of BAO and SNe Ia are low, however, and the CMB is only a snapshot of cosmic expansion. Some parameters, such as the density and EOS parameter of dark energy (Wang 2012; Wang & Dai 2014; Wang & Wang 2014b), might evolve with redshift. GRBs can probe the evolution of these parameters at high redshifts and serve as complementary tools for SNe Ia. The study of these evolutions can differentiate dark energy models. Some luminosity correlations have been proposed to standardize GRBs (Amati et al. 2002; Ghirlanda et al. 2004a; Liang & Zhang 2005). Ghirlanda et al. (2004a) found a tight correlation between collimated energy Eγ and the peak energy Ep of νFν spectrum. Dai et al. (2004) used this correlation to constrain cosmological parameters with 12 GRBs. Liang & Zhang (2005) found the EisoEptb correlation and used this correlation to constrain cosmological parameters. Recently, Wang et al. (2011b) constrained cosmological parameters with 109 GRBs using six GRB empirical correlations, and found in the flat Λ cold dark matter (CDM) model. Other attempts have also been made to standardize GRBs (Ghirlanda et al. 2004b; Friedman & Bloom 2005; Schaefer 2007; Wang et al. 2007; Liang et al. 2008; Kodama et al. 2008; Qi et al. 2009; Cardone et al. 2010; Wang & Dai 2011c). These methods of standardizing the long GRBs are mainly based on some empirical correlations, such as the EisoEp (Amati et al. 2002), EpLp (Schaefer 2003; Wei & Gao 2003), and EpEγ (Ghirlanda et al. 2004a), where Lp is the peak luminosity, Ep is the peak energy in cosmological rest frame, Eiso is the isotropic-equivalent energy, and Eγ is the collimation-corrected energy. Correlations within X-ray afterglow light curves have also been studied (Dainotti et al. 2008, 2010; Qi & Lu 2010).

In this paper, we focus on the usage of the EisoEp correlation. Amati et al. (2002) discovered this correlation with a small sample of BeppoSAX GRBs. Since many more GRBs are detected, attempts have been made to use this correlation for the purpose of cosmology. Amati et al. (2008) used a simultaneous fitting method to constrain the EisoEp correlation coefficients and cosmological parameters together with 70 long GRBs. The extrinsic scatter σext was taken into consideration in this method (D’Agostini 2005). Amati et al. (2008) assigned σext to Ep and found 0.04 < Ωm< 0.40 and σext = 0.17 ± 0.02 at 1σ confidence level in the flat ΛCDM Universe. For non-flat ΛCDM model, the results are Ωm ∈ [ 0.04,0.40 ] and ΩΛ< 1.05 (Amati et al. 2008). However, Ghirlanda (2009) doubted this result. He claimed that the extrinsic scatter term should be assigned to Eiso. This is consistent with D’Agostini (2005), who described that the extrinsic scatter σext should be assigned to the parameter that also depends on hidden variables (cosmological parameters in our study). We discuss this point in detail in Sect. 3.1. However, this would lead to no constraint on cosmological parameters with the same 70 GRBs from Amati et al. (2008). We test it again with a larger sample in this paper.

The calibration method is also helpful to standardize GRBs. Imitating the example of standardizing the standard candle of SNe Ia with Cepheid variables, GRBs can also be calibrated with SNe Ia (Liang et al. 2008; Kodama et al. 2008; Wei 2010; Lin et al. 2016). This method is also cosmological model independent. Liang et al. (2008) calibrated 42 high redshift GRBs with SNe Ia. Five interpolation methods were used and the results were consistent with each other. Wei (2010) standardized 59 high-redshift GRBs with SNe Ia, using the EisoEp correlation, and found that GRBs can improve the constraint on cosmological parameters. Wang & Dai (2011c) calibrated 116 GRBs with Union 2 SNe Ia with cosmographic parameters.

We use 151 GRBs, 109 of which are taken from Amati et al. (2008) and Amati et al. (2009). The remaining 42 GRBs are the updated long GRBs, which were detected by Fermi GBM, Konus-Wind, Swift-BAT, and Suzaku-WAM. The energy band, fluence, low (α), high (β) energy photon indices, spectral peak energy, and redshift are taken from the refined analysis of the corresponding GRB team. We test whether this larger GRB sample can help to constrain cosmological models better. First, we constrain the cosmological parameters and coefficients of the EisoEp correlation simultaneously. Then, we calibrate these GRBs with SNe Ia using the EisoEp correlation. At last, we compare these two methods and discuss them.

This paper is organized as follows. In the next section, we introduce the GRBs data and perform the K-correction. In Sect. 3, we test whether the redshift evolution of the EisoEp correlation is significant, and use a simultaneous fitting method to constrain cosmological parameters and coefficients of the EisoEp correlation. In Sect. 4, we use SNe Ia to calibrate the EisoEp correlation, then we use these calibrated GRBs to constrain cosmological parameters. Summary and discussions are given in Sect. 5.

Table 1

42 updated long GRBs.

2. Updated GRB sample

We collect all GRBs with information of redshift, fluence, peak energy, and photon indices from GCN Circulars Archive1Cucchiara et al. (2011) and Gendre et al. (2013) until February 13, 2014. The updated sample contains 42 updated long GRBs. We list these GRBs in Table 1. The spectra of these GRBs are obtained from the refined analysis of Fermi GBM team, Konus-Wind team, Swift-BAT team, and Suzaku-WAM team. The redshifts extend from 0.34 to 5.91. The spectrum is modeled by a broken power law (Band et al. 1993), (1)where Ep,obs is the observed peak energy, α and β are the low and high energy photon indices, respectively. We take the typical spectral index values for those GRB whose indices are not given out in the references, i.e., α = −1.0 and β = −2.2 (Salvaterra et al. 2009).

With these spectra parameters, we can obtain the peak energy in the cosmological rest frame by Ep = Ep,obs × (1 + z) and the bolometric fluence in the band of 1−104 keV by (Bloom et al. 2001) (2)where S is the observed fluence, Emin and Emax are the detection limits of the instrument, and z is the redshift.

In the EisoEp plane, Ep is an observed value, which is not dependent on the cosmological model. However, Eiso depends on the cosmological model from (3)where dL is the luminosity distance. Assuming a flat ΛCDM model, the dL can be expressed with Hubble expansion rate (4)where Ωm is the matter density at present, and H0 is the Hubble constant. Since the Hubble constant is precisely measured, we take H0 = 67.8 km s-1 Mpc-1 (Planck Collaboration XVI 2013; Planck Collaboration XIII 2015), except when we use the combination data of SNe and GRB to constrain cosmological models.

We list 42 updated GRBs in Table 1. The isotropic energy Eiso is calculated with benchmark parameters with Ωm = 0.308 for the flat ΛCDM Universe (Planck Collaboration XVI 2013; Planck Collaboration XIII 2015). During the calculation, we only take the errors propagating from the spectrum parameters, namely observed fluence S and peak energy Ep,obs. The uncertainties from other parameters are attributed into the extrinsic scatter σext.

3. The EisoEp correlation and constraints on cosmological parameters

3.1. The EisoEp correlation

To constrain cosmological models more precisely, we combine our updated 42 GRBs with 109 GRBs from Amati et al. (2008) and Amati et al. (2009). The full sample contains 151 GRBs and covers the redshift range from 0.0331 to 8.2. We parameterize the EisoEp correlation as follows: (5)where a and b are the intercept and slope. Here Ep has been corrected into the cosmological rest frame.

Table 2

EisoEp correlation fitting results of full data and four redshift bins.

Before constraining cosmological models, we test the possible redshift evolution of the EisoEp correlation using the maximum likelihood method. The full data is divided into four redshift bins: [ 0.0331,0.958 ], [ 0.966,1.613 ], [ 1.619,2.671 ] , and [ 2.69,8.2 ]. Each bin almost includes the same number of GRBs. The results are shown in Table 2. We give out the best-fit values and 1σ uncertainties in the coefficients a, b, and the extrinsic scatter σext. The σext is almost constant in different bins. Its value is about 0.34, which implies that the extrinsic scatter dominates the error size. The results show no statistically significant evidence for the redshift evolution of the EisoEp correlation. This result is consistent with those of Basilakos & Perivolaropoulos (2008) and Wang et al. (2011b). The full data result are also shown in Figure 1. This result illustrates that the EisoEp correlation fits the data well.

As discussed by D’Agostini (2005), we use the following likelihood to fit the linear relation y = a + bx, (6)Following the description of D’Agostini (2005), the parameter y should not only depend on x, but also depend on some hidden variables (Ωm here). Thus, the expression of the EisoEp plane should be written as and . However, Amati et al. (2008) set , thus the extrinsic scatter σext does not contain the error from the cosmological models.

thumbnail Fig. 1

EisoEp correlation. The solid black, dotted, and dashed lines represent the best-fit line, 1σext region, and 2σext region, respectively.

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3.2. Simultaneous fitting

Since the EisoEp correlation does not evolve with redshift, it can be used to constrain parameters directly. We emphasize that there is no circularity problem in the simultaneous fitting method because we do not assume any cosmological model. In this section, we focus on the constraint on the flat ΛCDM model. The luminosity distance is expressed as Eq. (4).

thumbnail Fig. 2

Evolution of log (ℒ)/log (ℒ)min as a function of Ωm in the flat ΛCDM Universe is shown with solid line from maximum likelihood method. The dashed line is the plot from reduced χ2 method. The dotted line is obtained with the 90 GRBs calibrated on the SNe Ia (see Sect. 4).

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Table 3

Constraints of cosmological parameters by GRBs.

Using the likelihood expressed in equation (6), we can constrain the current matter density Ωm, the extrinsic scatter parameter σext, and the coefficients of the EisoEp correlation simultaneously. In our calculations, the best-fit values are a = 49.15 ± 0.26, b = 1.42 ± 0.12, σext = 0.34 ± 0.03, and Ωm = 0.76. We show the constraint on Ωm in Fig. 2 with a solid line. The 1σ uncertainty is Ωm ∈ [ 0.55,1 ]. We also use the reduced χ2 method to constrain the matter density. This method also includes the effect of extrinsic scatter (7)The hidden variables (cosmological parameters) are included in Eiso. The extrinsic scatter is used to set the reduced χ2 to unity, which is also used in SNe Ia cosmology (Suzuki et al. 2012). The value of σext is 0.34 when the reduced χ2 is unity. The best-fit results are a = 48.96 ± 0.18, b = 1.52 ± 0.08, and Ωm = 0.50 ± 0.12. The constraint from reduced χ2 method is roughly consistent with the likelihood method. The evolution with Ωm are shown in Fig. 2 with a dashed line. If the extrinsic scatter is not considered, the results are a = 48.50 ± 0.05, b = 1.81 ± 0.02, and Ωm = 0.19 ± 0.05.

There is a mild tension between the results from the likelihood method and the reduced χ2 method. The extrinsic scatter is large, which loosely constrains the cosmological parameters. When we calculate the parameter Eiso, a cosmological model and a spectrum model are used, while the uncertainties from them are not well established, thus we take these uncertainties into a scatter parameter σext. This scatter should be assigned to the parameter Eiso. In the future, this scatter can be reduced, since precise observation and data analysis will be performed by the team of Sino-French space-based multiband astronomical variable objects monitor (SVOM; Basa et al. 2008; Götz et al. 2009; Paul et al. 2011).

We also compare our results to the current precise measurements, such as the results from Planck+WMAP (Planck Collaboration XVI 2013), BAO (Beutler et al. 2011; Anderson et al. 2014; Kazin et al. 2014; Ross et al. 2015), and SNe Ia (Conley et al. 2011; Suzuki et al. 2012). We show them in Table 3. The best-fit Ωm by GRBs, using χ2 method, conflicts with the observation of CMB and BAO. For the results from SNe Ia, however, if both statistical and systematic errors are included, the constraints on cosmological parameters are loose(Kowalski et al. 2008; Amanullah et al. 2010; Suzuki et al. 2012). In this case, the best-fit Ωm with GRBs, using χ2 method, is consistent with those from SNe Ia at 1σ confidence level; see Fig. 12 of Kowalski et al. (2008), Fig. 10 of Amanullah et al. (2010) and Fig. 5 of Suzuki et al. (2012).

Table 4

90 calibrated GRBs with redshift, bolometric fluence, peak energy in cosmological rest frame, and distance moduli.

4. Calibration of the EisoEp correlation

4.1. Standardizing GRBs with SNe Ia

Just as using Cepheid variables to standardize SNe Ia, the GRBs can be calibrated with SNe Ia. We can use the calibrating method to standardize the GRBs with the EisoEp correlation. With this approach, the parameters a and b are obtained and only cosmological parameters remain free. We use the latest Union 2.1 data from Suzuki et al. (2012). This method is also cosmological model independent (Liang et al. 2008; Kodama et al. 2008; Wei 2010). The extrinsic scatter is also be taken into account when calculating the error propagation of Eiso. The full GRB data is separated into two groups. The dividing line is the highest redshift in SNe Ia Union 2.1 data, namely, z = 1.414. The low-redshift group (z< 1.414) includes 61 GRBs and the high-redshift group (z> 1.414) contains 90 GRBs.

Firstly, the linear interpolation method is used to calibrate the distance moduli μ of 61 low-redshift GRBs. Liang et al. (2008) have shown that there are no differences on the final result between the linear interpolation and the cubic interpolation. The 1σ error of the distance moduli σμ,i can be obtained as follows:(8)where zi + 1 and zi are the redshift of the two nearest SNe Ia and ϵμ,i + 1 and ϵμ,i are the errors of these two SNe Ia. The redshift of interpolated GRB lies between zi and zi + 1.

After the distance moduli of 61 low-redshift GRBs are obtained, the luminosity distance can be derived from (9)Then the isotropic-equivalent energy Eiso can be calculated from Eq. (3). Following Schaefer (2007) and Liang et al. (2008), we use the bisector of the two ordinary least squares method (Isobe et al. 1990) to fit the EisoEp correlation. The best-fit values are a = 48.46 ± 0.033 and b = 1.766 ± 0.007. The result is shown in Fig. 3. The errors of distance moduli are not taken into consideration because the extrinsic scatter σext dominates the error size in the regression analysis (Schaefer 2007). Thus, we take σext directly into account during the calculations of the uncertainties of high-redshift GRBs (σlog Eiso). From the previous section, the value of σext is nearly constant, so we typically set σext = 0.34.

thumbnail Fig. 3

Low-redshift GRM sample EisoEp correlation. Black line is the best-fit result obtained by using the bisector of the two ordinary least squares method. The dotted line represents the 1 σext region and dashed line the 2 σext region.

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We have shown that the EisoEp correlation does not evolve with redshift in the previous section. Thus, the calibrated EisoEp correlation can be extrapolated to the high-redshift sample, namely, z> 1.414 group. Using Eq. (5), we can derive Eiso of high-redshift GRBs. The propagated uncertainties of Eiso can be calculated from (10)where the value of σext is 0.34. The values of σa and σb are derived from the bisector of the two ordinary least squares method.

Then, we use Eqs. (3) and (9) to derive the distance moduli. The propagated uncertainty is given by the following equation: (11)The calibrated 90 high-redshift GRBs are listed in Table 4. This sample can be used to constrain cosmological models directly. Compared with Wei (2010), the error bars of distance moduli of our results are smaller. The main reason is that we use a larger sample, which leads to a smaller σext. The extrinsic scatter parameter has been taken into consideration during the calculation of the error size of Eiso.

4.2. Constraining cosmological models

These GRBs carry the information of high-redshift Universe, and can be taken as good complements to the Union 2.1 data set. We test if these high-redshift GRBs alone can constrain the ΛCDM model. Using the distance modulus in Eq. (4) and Eq. (9), the χ2 is (12)where μcal is the calibrated GRB distance modulus listed in Table 4. The best-fit result is with 1σ uncertainty. The χ2 evolution with Ωm is shown in Fig. 2. This result is consistent with the constraints from SNe Ia (Conley et al. 2011; Suzuki et al. 2012), CMB (Planck Collaboration XVI 2013; Planck Collaboration XIII 2015), and BAO (Beutler et al. 2011; Anderson et al. 2014; Kazin et al. 2014; Ross et al. 2015) at 1σ confidence level, as shown in Table 3.

Since this GRB sample can constrain cosmological parameters successfully, we also combine the calibrated GRB data with SNe Ia from Union 2.1 sample to constrain cosmological models. For the flat ΛCDM, we obtain Ωm = 0.271 ± 0.019 and h = 0.701 ± 0.002, where h is the Hubble constant in units of 100 km s-1 Mpc-1. This is very consistent with the Union 2.1 SNe Ia data. For the non-flat ΛCDM, the luminosity distance is different and can be expressed as follows: (13)where (14)and (15)The χ2 of SNe Ia is constructed as follows: (16)Then the total χ2 is (17)The best-fit values with 1σ uncertainties are Ωm = 0.225 ± 0.044, ΩΛ = 0.640 ± 0.082, and h = 0.698 ± 0.004 for the combined sample (SNe+GRB). For the GRB sample, we obtain Ωm = 0.18 ± 0.11 and ΩΛ = 0.46 ± 0.51, which is consistent with the SNe Ia results at 1σ confidence level. The combined sample can help to constrain cosmological parameters much tighter because not only is the sample enlarged, but also the redshift covers a much wider. The flatness of the Universe depends on the curvature parameter, that is to say, Ωk = 1 − ΩΛ − Ωm. In Fig. 4, we use three samples, GRB, SNe, and combination of GRB+SNe to constrain the cosmological model. Both results prefer a flat Universe at the 1σ confidence level. The constraint from the GRB is almost perpendicular to that from SNe Ia in the Ωm − ΩΛ plane. Thus GRBs can significantly help to constrain Ωm because, in this redshift domain, the dark matter dominates the evolution of the Universe. We also show constraints on Ωmh in Fig. 5, and ΩΛh in Fig. 6.

thumbnail Fig. 4

1σ and 2σ constraints on Ωm and ΩΛ. We use three samples and plot them into different colors. The solid line shows the Ωk = 0 case.

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thumbnail Fig. 5

1σ and 2σ constraints on Ωm and h from SNe Ia and GRB data.

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thumbnail Fig. 6

1σ and 2σ constraints on the ΩΛ and h from SNe Ia and GRB data.

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5. Discussions and summary

In this paper, we update 42 long GRBs for the EisoEp correlation and combine them with 109 long GRBs from Amati et al. (2008) and Amati et al. (2009). This sample contains GRBs detected by different detectors with different sensitivities. Thus, the sample might be biased, but this bias should only have a weak effect on our results. We also use the complete sample to perform our analysis. We use the same criteria as Salvaterra et al. (2012) and Pescalli et al. (2015) to collect GRBs. The results are a = 49.45 ± 0.61, b = 1.24 ± 0.22 and σext = 0.38 ± 0.06, while no constraint on Ωm is found. These results are in tension with that of our updated full sample with a larger extrinsic scatter. No statistical evidence for the redshift evolution of the EisoEp is found in the full sample.

For cosmological purposes, we fit the EisoEp plane and the cosmological parameters simultaneously. Using a likelihood function we obtain a = 49.15 ± 0.26, b = 1.42 ± 0.11, σext = 0.34 ± 0.03, and Ωm ∈ [ 0.55,1 ]. Using the reduced χ2, we obtain a = 48.96 ± 0.18, b = 1.52 ± 0.08, and Ωm = 0.50 ± 0.12. The results from these two fitting methods are in mild tension. The main reason is that the extrinsic scatter of this correlation is too large. Thus, Ghirlanda (2009) finds no constraint with a smaller sample using the likelihood method. We also use a calibrating method. Based on the SNe Ia data, we obtain 90 calibrated GRBs. From these calibrated GRBs, we acquire for flat ΛCDM and for the non-flat ΛCDM, we obtain Ωm = 0.18 ± 0.11 and ΩΛ = 0.46 ± 0.51. We also combine the GRB sample with SNe Ia Union 2.1 data and obtain Ωm = 0.271 ± 0.019 and h = 0.701 ± 0.002 for the flat ΛCDM. For the non-flat ΛCDM, the results are Ωm = 0.225 ± 0.044, ΩΛ = 0.640 ± 0.082, and h = 0.698 ± 0.004. We list our results in Table 3, and compare them with the results from other current measurements. The results from GRBs are consistent with results from SNe Ia in 1σ confidence level (Conley et al. 2011; Suzuki et al. 2012), while they conflict with CMB (Planck Collaboration XVI 2013; Planck Collaboration XIII 2015) and BAO (Beutler et al. 2011; Anderson et al. 2014; Kazin et al. 2014; Ross et al. 2015). We also found that the GRBs can help to constrain dark matter better. The constraint from GRB are almost perpendicular to that from SNe Ia in the Ωm − ΩΛ plane. The main reason might be that at high redshift, the dark matter dominates the Universe.

The extrinsic scatter is taken into account in both the simultaneous fitting method and the calibrating method. Our results shows that tighter constraints on cosmological model can be obtained with the calibrating method. For the simultaneous fitting method, the reduced χ2 method gives a more stringent constraint on cosmological parameters than the likelihood method, but the constraint is still loose because of the large extrinsic scatter. This scatter is introduced by both cosmological models and the GRB spectrum parameters, such as Ep, fluence, and photon index. The spectrum parameters can be precisely measured by SVOM (Basa et al. 2008; Götz et al. 2009; Paul et al. 2011), which can reduce the extrinsic scatter. The GRBs from SVOM would better help shed light on the properties of early Universe.


Acknowledgments

We thank an anonymous referee for useful suggestions and comments. This work is supported by the National Basic Research Program of China (973 Program, grant No. 2014CB845800), the National Natural Science Foundation of China (grants 11422325, 11373022, 11103007, and 11033002), the Excellent Youth Foundation of Jiangsu Province (BK20140016), and the Program for New Century Excellent Talents in University (grant No. NCET-13-0279). K.S.C. is supported by the CRF Grants of the Government of the Hong Kong SAR under HUKST4/CRF/13G.

References

All Tables

Table 1

42 updated long GRBs.

Table 2

EisoEp correlation fitting results of full data and four redshift bins.

Table 3

Constraints of cosmological parameters by GRBs.

Table 4

90 calibrated GRBs with redshift, bolometric fluence, peak energy in cosmological rest frame, and distance moduli.

All Figures

thumbnail Fig. 1

EisoEp correlation. The solid black, dotted, and dashed lines represent the best-fit line, 1σext region, and 2σext region, respectively.

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In the text
thumbnail Fig. 2

Evolution of log (ℒ)/log (ℒ)min as a function of Ωm in the flat ΛCDM Universe is shown with solid line from maximum likelihood method. The dashed line is the plot from reduced χ2 method. The dotted line is obtained with the 90 GRBs calibrated on the SNe Ia (see Sect. 4).

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In the text
thumbnail Fig. 3

Low-redshift GRM sample EisoEp correlation. Black line is the best-fit result obtained by using the bisector of the two ordinary least squares method. The dotted line represents the 1 σext region and dashed line the 2 σext region.

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In the text
thumbnail Fig. 4

1σ and 2σ constraints on Ωm and ΩΛ. We use three samples and plot them into different colors. The solid line shows the Ωk = 0 case.

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In the text
thumbnail Fig. 5

1σ and 2σ constraints on Ωm and h from SNe Ia and GRB data.

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In the text
thumbnail Fig. 6

1σ and 2σ constraints on the ΩΛ and h from SNe Ia and GRB data.

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In the text

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