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Effect of GRB spectra on the empirical luminosity correlations and the GRB Hubble diagram

The spectra of gamma-ray bursts (GRBs) in a wide energy range can usually be well described by the Band function, which is a two smoothly jointed power laws cutting at a breaking energy. Below the breaking energy, the Band function reduces to a cut-off power law, while above the breaking energy it is a simple power law. However, for some detectors (such as the Swift-BAT) whose working energy is well below or just near the breaking energy, the observed spectra can be fitted to cut-off power law with enough precision. Besides, since the energy band of Swift-BAT is very narrow, the spectra of most GRBs can be fitted well even using a simple power law. In this paper, with the most up-to-date sample of Swift-BAT GRBs, we study the effect of different spectral models on the empirical luminosity correlations, and further investigate the effect on the reconstruction of GRB Hubble diagram. We mainly focus on two luminosity correlations, i.e., the Amati relation and Yonetoku relation. We calculate these two luminosity correlations on both the case that the GRB spectra are modeled by Band function and cut-off power law. It is found that both luminosity correlations only moderately depend on the choice of GRB spectra. Monte Carlo simulations show that Amati relation is insensitive to the high-energy power-law index of the Band function. As a result, the GRB Hubble diagram calibrated using luminosity correlations is almost independent on the GRB spectra.


INTRODUCTION
Gamma-ray bursts (GRBs), mainly emitting radiation in the gamma-ray waveband which lasts a few seconds, and may be followed by the X-ray, optical or radio emissions lasting a few days, are the most energetic explosions in the Universe (Piran 1999;Mészáros 2006;Kumar & Zhang 2015). The recent decades have seen great successes in understanding the nature of GRBs thanks to the contribution by various space experiments. The isotropic distribution in the sky strongly implies that GRBs happen in the deep universe (Paciesas et al. 1999). The cosmological origin of GRBs was eventually confirmed starting form 1997 thanks to the combination of accurate localizations by BeppoSAX and redshift measurements from the follow-up observation with the main optical/IR facilities. The furthest GRB observed at present has redshift ∼ 9.4 (Cucchiara et al. 2011), and the total isotropic equivalent radiated energy by a GRB is about 10 48 ∼ 10 55 ergs (Kumar & Zhang 2015). The durations of GRBs span about six orders of magnitude, from milliseconds for the shortest to thousands of seconds for the longest. It is widely accepted that GRBs can be divided into two classes according to the duration, because the histogram of duration is fitted well by the summation of two Gauss functions separated at around 2 seconds (Kouveliotou et al. 1993). Therefore, short and long GRBs are classified depending on whether the duration is shorter or longer than 2 seconds. The light curves of GRBs have various type, from smoothly decaying to highly variable with many spikes (Fishman & Meegan 1995). The spectrum of a typical GRB can be well fitted by the Band function (Band et al. 1993), which is a two smoothly jointed power lows cutting at a breaking energy. Below the breaking energy, the Band function reduces to the cut-off power law (CPL), and above the breaking energy, it is the simple power law (PL). However, some detectors such as the Swift-BAT, work in an energy band well below or just near the breaking energy. Therefore, it is enough to fit the spectra just with the simple PL, and the high-energy powerlaw index of the Band function couldn't be constrained.
Since the first linear correlation between the spectral lag and the isotropic peak luminosity (τ lag − L relation) was found by Norris, Marani & Bonnell (2000), several other empirical correlations have been found. These empirical correlations relate the luminosity to the spectral parameters of GRBs. One of the most famous correlation was found by Amati et al. (2002). The Amati relation is a linear (in the logarithmic scale) correlation between the isotropic equivalent energy and the peak energy of νFν spectra (Ep − Eiso relation). Later, Yonetoku et al. (2004) found a similar correlation between the isotropic peak luminosity and the peak energy (Ep − L relation). Other two-parameter luminosity correlations includes: V − L relation (Fenimore & Ramirez-Ruiz 2000), Ep − Eγ relation , τRT − L relation (Schaefer 2007), and so on. Besides, there are also three-parameter correlations such as Eiso − Ep − t b relation (Liang & Zhang 2005), L − Ep − T0.45 relation (Firmani et al. 2006), EX,iso − Eγ,iso − Ep relation (Bernardini et al. 2012;Margutti et al. 2013), L−Ep −Γ0 relation (Liang et al. 2015). Some luminosity correlations are only valid for long GRBs, while others are valid for both short and long GRBs. These correlations are often used to calibrate GRBs as the standard candles and to reconstruct the GRB Hubble diagram (Schaefer 2003;Dai, Liang & Xu 2004;Liang & Zhang 2005;Firmani et al. 2005;Schaefer 2007;Amati 2008;Basilakos & Perivolaropoulos 2008;Wei & Zhang 2009;Wei 2010;Liu & Wei 2014;Wang et al. 2015;. The above luminosity correlations involve three quantities, i.e., the isotropic equivalent energy Eiso, the isotropic peak luminosity L and the spectral peak energy Ep. The derivation of the first two quantities involves the integration of spectra over the " bolometric" 1 − 10000 keV energy band. However, most detectors such as Swift-BAT only measure photons in a narrow energy band. Therefore, we should extrapolate the observed spectra to the whole energy band. Most previous works calculated Eiso and L by assuming that the spectra are modeled by the Band function. Due to the narrow energy width of detectors, CPL (or even PL) is enough to fit the observed spectra, and the high-energy index (β) of the Band function can't be constrained. Schaefer (2007) fixed β to be −2.2 for all the GRBs that have no measured β. In fact, about 60% (41 out of 69) GRBs compiled in Schaefer (2007) have no measurement of β. The following works Basilakos & Perivolaropoulos 2008;Xiao & Schaefer 2009;Wei 2010;Wang, Qi & Dai 2011;Liu & Wei 2014) which cited the data from Schaefer (2007) also have the underling assumption that the unmeasured β's are all set to be −2.2. It is especially necessary to check such a setting has some influence on the final luminosity correlations. Using 29 Swift GRBs, Cabrera et al. (2007) found that the zero-point of Amati relation calculated from CPL is slightly larger than that from Band function, while the intercept does not change significantly. The small change of zero-point of Amati relation may have some influences on the calibration of GRBs.
In this paper, we investigate if the luminosity correlations are affected by the choice of GRB spectra. We search the most up-to-date Swift data archive 1 and find that among ∼ 300 GRBs which have redshift measurement, only 44 GRBs can be well fitted by CPL so have well defined peak energy. Using the 44 GRBs, we study the Ep − Eiso correlation and Ep − L correlation. All the 44 GRBs are fitted by CPL so the high-energy index couldn't be determined. Therefore, we calculate Eiso and L by using the CPL instead of the Band function. To comparison, we also calculate Eiso and L using the Band function, with the high-energy index fixed to be −2.2, the median value of BATSE GRBs (Preece et al. 2000), as was done by Schaefer (2007). We further calibrate the distance of GRBs using these two luminosity correlations respectively, and reconstruct GRB Hubble diagram. We investigate the influence of different spectra on the luminosity correlations and on the reconstruction of GRB Hubble diagram. Throughout this paper, we assume the concordance cosmological model with the fiducial parameters H0 = 70 km s −1 Mpc −1 , ΩM = 0.28 and ΩΛ = 0.72.
The rest of this paper is arranged as follows: In Section 2, we present the observational properties of 44 Swift-BAT GRBs used in our analysis. In Section 3, we investigate the Ep − Eiso relation and Ep − L relation by assuming that the GRB spectrum is modeled by Band function and CPL, respectively, and compare their differences. We also implement Monte Carlo simulations to test the sensitivity of luminosity correlations on the high-energy power-law index. In Section 4, we calibrate the distance of GRBs through these two luminosity correlations, to see whether the resulting GRB Hubble diagram would depend on the choice of spectra. Finally, Section 5 is devoted to a short summary.

THE OBSERVATIONAL PROPERTIES OF SWIFT GRBS
The Swift satellite, launched on November 20, 2004, is a first-of-its-kind multi-wavelength observatory dedicated to the study of GRB science. There are three instruments onboard Swift working together to observe GRBs and afterglows in the gamma-ray, X-ray, ultraviolet, and optical wavebands. The Burst Alert Telescope (BAT, 15−150 keV), which is a highly sensitive and large field of view gamma-ray detector working in the energy band 15 − 150 keV, detects GRBs and accurately determines their positions on the sky. Soon after the triggering of BAT, the X-ray Telescope (XRT, 0.3 − 10 keV) and UV/Optical Telescope (UVOT, 170 − 600 nm) quickly repoint the field of view towards GRBs and start the afterglow observation. With the rate of about 100 GRBs per year, Swift has observed about one thousand GRBs until September, 2015, and about 300 GRBs have redshift measurement.
For most GRBs, the spectra can be well fitted by a two smoothly jointed power laws cutting at a breaking energy E b , which is well known as the Band function (Band et al.

1993),
(1) where α and β are the low-energy and high-energy powerlaw index, respectively, and they vary from bursts to bursts. Statistically, α peaks around −1, and β peaks around −2.2 (Preece et al. 2000). The Band function is continuous and smooth at the breaking energy E b ≡ (α − β)/(2 + α)Ep, and it is monotonously decreasing in the whole energy range. However, it is easy to prove that the νFν ∝ E 2 N (E) spectrum has a peak at E = Ep, for any α −2 and β < −2. The distribution of Ep depends on the specific detectors. For the BATSE catalog, Ep peaks around 150 keV (Preece et al. 2000), while for the Swift-BAT catalog, Ep is relatively lower (Sakamoto et al. 2011). The Band function was first proposed from analyzing the spectra of GRBs observed by BATSE, which is a gamma-ray detector sensitive in a wide energy band 20 − 2000 keV. However, the Swift-BAT is only sensitive in a low and narrow energy band, which is just near or even below the typical breaking energy. Therefore, the observed spectra by Swift-BAT can be fitted using the low-energy end of Band function, i.e., the CPL, This is to say, the high-energy index β can't be well constrained. The CPL is a special case of Band function in the limit β → ∞. Figure 1 shows the difference between the Band function and CPL. CPL falls much more dramatically than Band function above E b . On the other hand, since the Swift-BAT working energy band is very narrow, for majority of GRBs we can even fit the spectra just with a simple PL, i.e., The νFν spectrum of CPL also peaks at Ep, while PL is monotonous and has no peak. Among ∼ 300 Swift GRBs with measured redshift, only 44 GRBs have a reasonable CPL fit so have well determined peak energy, while the remaining GRBs can be fitted well by the simple PL. Only these 44 GRBs are available to analyze the luminosity correlations. We list the observational properties of the 44 GRBs in Table 1. The errors in this table are of 90% confidence level (C.L.). All of the 44 GRBs are long GRBs with duration T90 > 2 seconds 2 . The observed photon fluence spans three orders of magnitude, while the peak photon flux spans two orders of magnitude. Figure 2(a) is the histogram of redshift of the 44 GRBs. The redshift of our sample ranges from 0.347 to 9.4, with a median value 1.9. Figure 2(b) is the histogram of the low-energy power-law index. A Gauss fit to the histogram gives the average valuē α = −1.05 ± 0.14 and standard deviation σα = 0.38 ± 0.14 (95% C.L.). Two GRBs (GRB 131117A and GRB 130612A) have positive index. Figure 2(c) is the distribution of the observed peak energy Ep, ranging from 19.2 keV to 258 keV, with a median value 71.4 keV. Only 9% (4 out of 44) GRBs have peak energy above the BAT energy band. All the remaining GRBs have peak energy well within the BAT energy range. This is one reason why the high-energy power-law index can't be well constrained. In Figure 2(d) we plot the histogram of the rest frame peak energy Ep,i = Ep × (1 + z), which ranges from 38 keV to 693 keV. The distribution of Ep,i is more homogeneous compared to that of Ep .

THE LUMINOSITY CORRELATIONS
The most investigated GRB luminosity correlations are those between the total radiated energy or luminosity (calculated in a fixed energy band) and the spectral peak energy. Here we focus on discussing two luminosity correlations, i.e., the Amati relation (Ep − Eiso) and Yonetoku relation (Ep − L).
The Amati relation is a linear (in the logarithmic scale) correlation between isotropic equivalent energy and peak energy of νFν spectrum in the comoving frame (Amati et al. 2002;Amati 2003Amati , 2006. It was first discovered by Amati et al. (2002) in 12 long BeppoSAX GRBs and was confirmed in larger samples (Amati 2003(Amati , 2006. Later it was found that short GRBs do not follow the same Amati relation of long GRBs (Amati, Frontera & Guidorzi 2009;Amati 2010). The physical origin of Amati relation has been discussed but selection effect couldn't be excluded. Amati relation is one of the most used relations in calibrating the distance of GRBs, although there are still some controversies on its validity (Li Table 1. The observed properties of 44 Swift-BAT GRBs. Column (1): the GRB identifier. Columns (2) and (3): the Right Ascension and Declination (J2000) in unit of degree. Column (4): the GRB duration in which time 90% photons are recorded. Column (5): the redshift. Column (6): the observed photon fluence in 15 − 150 keV energy band in unit of 10 −7 erg cm −2 . Column (7): the observed peak photon flux in 15 − 150 keV energy band in unit of photons cm −2 s −1 . Column (8): the photon index. Column (9): the observed peak energy in unit of keV. Column (10): the GCN Circular numbers (http://gcn.gsfc.nasa.gov/gcn3¯archive.html). All the errors in this table are of 90% C.L. Note that in the Swift data archive, the uncertainties of Ep (and/or S, P , α) of some GRBs are not symmetric. We symmetrize them by taking the average, i.e., σ = (σ − + σ + )/2. 2007; Dainotti et al. 2013;. We can parameterize Amati relation as where Ep,i = Ep × (1 + z) is the peak energy in the rest frame, and is the isotropic equivalent energy in the rest frame 1−10, 000 keV energy band, and dL is the luminosity distance. In the concordance cosmological model, the luminosity distance is given by . (6) The bolometric fluence S bolo is calculated from the observed photon fluence S and GRB spectra N (E) in the rest frame 1 − 10, 000 keV energy band (Schaefer 2007), where Emin = 15 keV and Emax = 150 keV are the lower and upper boundary of the BAT energy band, respectively. The bolometric fluence S bolo , as well as the isotropic equivalent energy Eiso, depend on the GRB spectra. We calculate S bolo and Eiso in both cases that GRB spectra are modeled by Band function and CPL. Since the high-energy power-law index of the Band function can't be measured, we fix it to be −2.2, the peak of BATSE catalog (Preece et al. 2000). The final results are listed in columns (2) − (5) in Table 2. Errors are given in 1σ C.L. We only consider the error propagation from S. As is expected, the isotropic equivalent energy calculated from Band function is much larger than that from CPL, since CPL falls much faster than Band function above the breaking energy. The Yonetoku relation is first found by Yonetoku et al. (2004) from the combination of 12 BeppoSAX GRBs and 11 BATSE GRBs. It is similar to the Amati relation except that the isotropic equivalent energy Eiso is replaced by the isotropic peak luminosity L, i.e., The isotropic peak luminosity L can be calculated from the bolometric peak flux P bolo as while the bolometric peak flux P bolo is calculated from the observed peak photon flux P in the rest frame 1−10, 000 keV energy band by integrating over the GRB spectra (Schaefer 2007), We also calculate P bolo and L in both the Band and CPL spectra cases, and list the results in columns (2) − (5) in Table 3. Errors are given in 1σ C.L. The errors of P bolo and L are propagated from the errors of P . Similarly, we can see that the peak luminosity calculated from Band function is much larger than that from CPL. In order to see more clearly the effect of GRB spectra on the final isotropic energy (and peak luminosity), we calculate the difference of isotropic energy (and peak luminosity) derived from Band and CPL spectra, We plot ∆Eiso and ∆L versus the power-law index α in Figure 3. We can see that both ∆Eiso and ∆L are positively correlated with α. When α is close to −2, both ∆Eiso and ∆L are small. This is because the difference between Band function and CPL model becomes smaller when α approaches β. Both Amati relation and Yonetoku relation have the  (2) and (3): the bolometric fluence in unit of 10 −7 erg/cm 2 . Columns (4) and (5): the isotropic equivalent energy in unit of 10 52 ergs. Columns (6) and (7): the distance modulus calibrated using the Amati relation. The superscripts " CPL" and " Band" mean that the quantities are calculated using the CPL and Band function, respectively. All the errors in this table are of 1σ C.L. (1) The best-fitting parameters (a, b) can be derived using the likelihood method presented in D'Agostini (2005), i.e., by maximizing the following likelihood: where σint is the intrinsic scatter, which represents the unknown errors. The introduction of intrinsic scatter is necessary since it dominates over the measurement error.
We plot the Amati relation and Yonetoku relation in Figure 4, and list the best-fitting parameters in Table 4. From Table 4, we can see that the intercept parameters of both Ep − Eiso relation and Ep − L relation calculated from the Band spectrum are moderately larger than that from CPL spectrum, while the slope parameters show no significant difference. This is consistent with the results of Cabrera Table 3. The bolometric peak flux, isotropic peak luminosity, and distance modulus calibrated using the Yonetoku relation. Column (1): the GRB identifier. Columns (2) and (3): the bolometric peak flux in unit of 10 −7 erg/cm 2 /s. Columns (4) and (5): the isotropic peak luminosity in unit of 10 51 erg/s. Columns (6) and (7): the distance modulus calibrated using the Yonetoku relation. The superscripts " CPL" and " Band" mean that the quantities are calculated using the CPL and Band function, respectively. All the errors in this table are of 1σ C.L. (1) (2) They found that a ′ in the Band case is smaller than in the CPL cases.
A smaller a ′ corresponds to a larger a in our parametrization.
tion are well consistent with the results of Schaefer (2007). However, the slope parameters of Amati relation are somehow smaller than previous results (Amati et al. 2002;Amati 2003Amati , 2006. One possible reason for the discrepancy may be that our sample is purely obtained from the Swift catalog, while previous works collected sample from other catalog (or the combination of various catalogs). For example, Amati (2006) calculated the Ep − Eiso relation of 41 GRBs from the combination of BATSE, BeppoSAX, HETE-2, Konus and Swift catalogs, and found a slope of ∼ 2. In the Ep −Eiso plot, the Swift subsample seems to fall near the best-fitting line of the full sample. Therefore, he claimed that the Swift subsample follows the same Ep − Eiso relation of the full sample. However, Amati (2006) has not calculated the slope parameter for the Swift subsample alone. In fact, if we only consider the 9 Swift GRBs corrected in table 1 of Amati's paper, we get a slope of ∼ 1.26, which is much flatter than that of the full sample. For the pure Swift GRBs collected in Table 13 of Sakamoto et al. (2011), we get a slope of ∼ 1.47, still much smaller than the originally proposed value ∼ 2 (Amati et al. 2002). This may imply that the Swift GRBs have a flatter Ep −Eiso slope than other GRBs. In the bottle two panels of Figure 4, we plot the 1σ, 2σ and 3σ contours in the (a, b) plane. Red contours represent the CPL case and black contours represent the Band case. The best-fitting values are denoted by dots. It seems that both Ep−Eiso relation and Ep −L relation in the Band cases are differs from that in the CPL case at about 2σ C.L. In the last column of Table  4, we also list the Pearson's linear correlation coefficient. We can see that the correlations in the CPL case are as tight as that in the Band case. However, both correlations are not as tight as previously claimed. The intrinsic scatters of both luminosity correlations are comparable to the results of Wang, Qi & Dai (2011).
Among the 44 Swift-BAT GRBs considered here, at least 24 of them are also observed by Konus-Wind (Aptekar et al. 1995) or Fermi-GBM (Meegan et al. 2009). These two detectors can observe GRBs in a much wider energy band (10 keV − 10 MeV for Konus-Wind and 8 keV − 40 MeV for Fermi-GBM) than Swift-BAT, making the detailed analysis of spectra possible. We list the spectral properties of these 24 GRBs in Table 5. The spectra of 10 GRBs can be fitted to the Band function, so that the highenergy index β can be well constrained. For the remain 14 GRBs, the spectra can only be fitted to CPL, and β still cann't be constrained. In column (8) of Table 5, we also list the isotropic-equivalent energy Eiso. For GRBs which have well measured β, Eiso is calculated from the Band model. Otherwise, Eiso is calculated from the CPL model. With these 24 GRBs, we calculate the Amati relation and obtain parameters a = 52.749 ± 0.143, b = 1.285 ± 0.538, σint = 0.701 ± 0.101. Due to the narrow coverage of Ep and the small sample, the intrinsic scatter is large and the slope parameter couldn't be well constrained. Nevertheless, the slope still seems to be flatter than previous findings, but consistent with our results. It is not clearly understood why Swift GRBs have flatter slope. One reason may be that the peak energies measured by Swift/BAT have a relatively narrow distribution (e.g., most of Ep,i's are between 50 and  (3): the observed photon fluence in unit of 10 −7 erg cm −2 . Column (4): the low-energy photon index. Column (5): the high-energy photon index. Column (6): the observed peak energy in unit of keV. Column (7): the energy band in which the photon fluence is calculated, in unit of keV. Column (8): the isotropic equivalent energy in unit of 10 52 ergs. Column (9): the detector by which the GRB was observed, KON for Konus-Wind and GBM for Fermi-GBM. Column (10): the GCN Circular numbers. All the errors in this table are of 90% C.L. (1) (2) 500 keV), while the samples investigated previously cover at least 3 orders of magnitude in Ep,i (from a few keV to a few thousands keV).
To test the sensitivity of luminosity correlations on the high-energy power-law index β, we implement the Monte Carlo simulations. We recalculate the Amati relation in the case that GRB spectra are modeled by the Band function. In this time, β is no longer fixed at −2.2, but it is a random number generated from the Gauss distribution with average value −2.2 and standard deviation 0.4 4 . We simulate 1000 times and derive the intercept and slope parameters (a, b) of Amati relation in each simulation. The histograms of a and b in 1000 simulations are plotted in Figure 5. The distribution of both a and b can be well fitted by Gauss function, with average values and standard deviations µa = 52.890 ± 0.001, σa = 0.010 ± 0.001 (95% C.L.), and µ b = 1.032 ± 0.010, σ b = 0.030 ± 0.002 (95% C.L.), respectively. This is consistent with the Amati relation in both the CPL and Band cases (see Table 4). The small standard deviations of a and b imply that the Amati relation is insensitive to β. Since the Yonetoku relation in the CPL cases does not much differ from that in the Band case, we can expect that it is also insensitive to β.

DISTANCE CALIBRATION AND GRB HUBBLE DIAGRAM
Both the Amati relation and Yonetoku relation have been widely used to calibrate the distance of GRBs. Here we take the Amati relation as an example. The calibrating procedures are as follows: first, derive the best-fitting parameters (a, b) of Amati relation as discussed in the last section. Second, calculate Eiso of each GRB through Eq.(4), by fixing (a, b) at the best-fitting values. Then the luminosity distance of each GRB can be obtained through Eq.(5). Finally, convert the luminosity distance to distance modulus, i.e., µ(z) = 5 log dL(z) Mpc + 25.
The uncertainty of distance modulus is calculated using the formulae of error propagation (Schaefer 2007),  Figure 6. The GRB Hubble diagram and residuals of distance moduli, calibrated using the Amati relation. Left two panels: in the case of CPL spectrum. Right two panels: in the case of Band spectrum. In the top two panels, the black curve is the theoretical result of concordance ΛCDM model (Ω M = 0.28, Ω Λ = 0.72, H 0 = 70 km s −1 Mpc −1 ). In the bottle two panels, the black (red) error bars represent that the calibrated distance moduli are consistent (inconsistent) with the ΛCDM model within 1σ uncertainty.
We can also calibrate GRBs using the Yonetoku relation in a very similar way.
We must point out that the calibrating method mentioned above has the circularity problem and is model dependent, because we have used the concordance cosmological model with fiducial parameters in the calibration. Our paper is not aimed at solving this problem, but we are rather interested in studying the effect of GRB spectra on the distance calibration. Some model-independent methods free of circularity problem have been extensively discussed Wei & Zhang 2009;Wei 2010;Liu & Wei 2014).
The distance moduli calibrated using the Amati relation and Yonetoku relation are listed in columns (6) − (7) in Table 2 and Table 3, respectively. We can see that the distance moduli only slightly depend on the choice of GRB spectra. The difference between distance moduli in the two spectra cases (Band and CPL) is much smaller than the uncertainties of distance moduli, i,e, |µ Band − µ CPL | ≪ σµ. In Figure 6, we plot the GRB Hubble diagram calibrated using Amati relation. The top-left and top-right panels are the resulting Hubble diagram in the CPL and Band cases, respectively. The solid curves are the theoretical results of concordance ΛCDM model with fiducial parameters H0 = 70 km s −1 Mpc −1 , ΩM = 0.28 and ΩΛ = 0.72. In the bottle two panels of Figure 6, we plot the residual of distance moduli with respect to the theoretical curve. The error bar represents 1σ uncertainty. GRBs consistent (inconsistent) with theoretical curve within 1σ uncertainty are denoted by black (red). Due to the large intrinsic scatter of Amati relation, 27% (12 out of 44) GRBs do not fall onto the theoretical curve within 1σ uncertainty. These GRBs all have low redshift (z 4). GRBs inconsistent with theoretical curve in the CPL case are still inconsistent in the Band case. The uncertainty of distance modulus mainly ( 90%) propagates from the intrinsic scatter. A much tighter correlation with smaller intrinsic scatter is necessary to reduce σµ.
We also reconstruct the GRB Hubble diagram using the Yonetoku relation and plot it in Figure 7. Compared to the Hubble diagram reconstructed from Amati relation, very similar conclusion can be draw. Therefore, we conclude that the GRB Hubble diagram almost does not depend on the choice of GRB spectra.

SUMMARY
In this paper, we have investigated the effect of GRB spectra on the luminosity correlations and GRB Hubble diagram. Most previous works calculated the isotropic equivalent energy and peak luminosity by assuming that GRB spectra are modeled by the Band function. Due to the narrow energy coverage of detectors, the high-energy power-law index of Band function usually can't be constrained. Using 44 Swift GRBs with well determined peak energy, we calculated two luminosity correlations (Amati relation and Yonetoku relation) in the cases that GRB spectra are modeled by the Band function and CPL, respectively. It is found that both Amati relation and Yonetoku relation only moderately depend on the choice of GRB spectra. Monte Carlo simulations show that the Amati relation is insensitive to the high-energy power-law index of Band function. We further calibrated the distance of GRBs using these two luminosity correlations and reconstructed the GRB Hubble diagram. We found that different GRB spectra only cause the difference of distance modulus at the order of 0.1 mag, while the uncertainty of distance modulus calibrated using luminosity correlations are usually larger than 1.0 mag. Therefore, we may conclude that the resulting GRB Hubble diagram is insensitive to the choice of GRB spectra.