1 Introduction

Gamma-ray bursts (GRBs) are sudden, and very brief, outbursts of high-energy gamma photons that appear randomly in time and space. They were serendipitously discovered in 1967, and are of great importance because they are currently the most luminous and distant sources in the universe. They hold great promise as cosmological probes of the early universe, since their flux is unencumbered by extinction due to dust (Ghirlanda et al. 2006; Azzam and Alothman 2006a, 2006b; Capozziello and Izzo 2008; Demianski et al. 2011). One of the most important elements in the detection of GRBs is the redshift, z, since its determination is necessary in order to investigate all the intrinsic characteristics of GRBs. It is generally determined by the identification of absorption lines in the optical afterglow spectra, when they are bright enough. Large terrestrial telescopes equipped with spectrographs working in the infrared or the optical domains are the best places to perform this task.

Over the past decade, several GRB energy and luminosity relations have been discovered, in which an observed parameter correlates with an intrinsic parameter. Some of these relations are obtained from the light curves, like the lag-luminosity relation (Norris et al. 2000) and the variability relation (Fenimore and Ramirez-Ruiz 2000), while others are extracted from the spectra and include the Amati relation (Amati et al. 2002, 2008, 2009; Amati and Mon 2006), the Ghirlanda relation (Ghirlanda et al. 2004), the Yonetoku relation (Yonetoku et al. 2004; Ghirlanda et al. 2010), and the Liang-Zhang relation (Liang and Zhang 2005).

In this work, we use a sample of Swift GRBs that we selected according to a specific criterion (described in Sect. 2) in order to investigate two of these correlations: the Amati relation, which is a correlation between the intrinsic (i.e., rest-frame) peak energy, E p,i , in a burst’s νF ν spectrum and its equivalent isotropic energy, E iso ; and the Yonetoku relation which is a correlation between E p,i and a burst’s isotropic peak luminosity, L iso .

A detailed description of our sample selection is provided in Sect. 2, which is followed by a presentation of our spectral analysis and results in Sects. 3 and 4, respectively. A discussion of our results including a comparison with what has been done by others is given in Sect. 5, and our conclusions are provided in Sect. 6.

2 Sample selection

We use the Swift GRBs data that is published on the official websites.Footnote 1 Footnote 2 The first one presents the observational results characterizing the overall GRB: peak flux, fluence, duration, redshift, host galaxy, as well as data on afterglows. The second website provides more details on the energy spectrum and the time profile in different energy bands, for different time resolutions. The data are all provided with their uncertainties (error bars). We simply use that data to check for the validity of the Amati and Yonetoku correlations. Bursts that interest us are therefore long-duration GRBs (LGRBs) with measured redshifts.

As of 25/09/2013, Swift observed 809 GRBs, of which 703 are long-duration GRBs. Among these 703 LGRBs, only 236 have measured redshifts, of which 17 are “approximate” redshifts (060708, 071020, 050904, 110726A, 100704A, 090814A, 070721B, 060912A, 070306, 050803, 120521C, 060116, 100728B, 081222, 060502A, 080430, and 050802), which leaves 219 LGRBs with well determined redshifts. We note that in cases where several methods for the determination of a redshift were possible, we have adopted the values given by the absorption method, which is generally the most precise (with four significant figures).

In Fig. 1 we plot the distribution of 219 LGRBs detected by Swift up to 25/09/2013. Of these, we select the bursts that have an energy \(E^{obs}_{p}\) in the Swift range [15–150] keV, with a relative accuracy of at least 70%. This selection filter leaves us with 71 GRBs. Among these, 57 have both L iso and E p;i values and thus can be used for testing the Amati relation, while 56 have E iso and E p;i values and thus can be used for testing the Yonetoku relation. 42 GRBs are common to both relations. The last constraint that we apply in selecting Swift bursts is the photon flux which must be more than the threshold P ph =2.6 ph/cm2/s. The sample we obtain is one of 27 “good” GRBs for our analysis. We note that these 27 bursts have been very strictly selected. The percentage of bursts that obey the observational constraints is of the order of 4 % of the total number of LGRBs, and it is 13 % of LGRBs with well-determined redshift.

Fig. 1
figure 1

GRB’s distribution with redshift z, with a bin of 0.2

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3 Spectral analysis

The 1-sec photon flux at the peak can be found in the Swift data. This flux gives the total number of photons per unit area and per unit time, regardless of their individual energies. The Swift energy spectrum is divided into four bands: 15–25 keV, 25–50 keV, 50–100 keV, 100–350 keV. The spectrum (photons per unit area, time, and energy) can be fitted by one of three well known functions: the Band function (Band et al. 1992), the cut-off power law (CPL), and the power law (PL). The first two are known to give very similar chi-square (χ 2) values when fitting the spectra. In the aforementioned Swift data web sites, only the PL and the CPL functions are given for each burst; thus, in this work, we have chosen to use the cut-off power law, which is characterized by two spectral parameters: the observed peak energy, \(E^{obs}_{p}\), and the spectral index alpha (α).

$$ N(E)=N_0\biggl(\frac{E}{E_n}\biggr)^{\alpha} e^{-E/E_0}, $$
(1)

where E n =50 keV, a mid-range value in the interval 15–150 keV which is only used for normalization purposes and \(E_{0} =E^{obs}_{p}/(2+\alpha)\). The spectral parameters \(E^{obs}_{p}\) and α in the CPL function are not necessarily the same as those of the Band law for a given burst. This observation has a direct effect on the two correlation relations since both of them depend on E p,i .

To calculate the peak energy flux, expressed in erg/cm2/s, we use the spectral parameters that characterize the peak photons. This is given under the heading “1-s peak spectral analysis” of the Swift data page. We only take what relates to the CPL spectrum. Using the CPL law, the observed peak fluxes P ph that are given in Table 4, are calculated theoretically using the following expression:

$$ P_{ph} =N_0\int_{E_{min}}^{E_{max}} \biggl(\frac{E}{E_n}\biggr)^{\alpha} e^{-E(2+\alpha)/E^{obs}_p} dE, $$
(2)

where E min =15 keV, E max =150 keV. P ph , α and \(E^{obs}_{p}\) being given in Table 4. In Eq. (2), the only unknown is the normalization constant N 0, which we determine by numerically integrating the previous function, i.e.

$$ N_0 =\frac{P_{ph}}{\int_{E_{min}}^{E_{max}}(\frac{E}{E_n})^{ \alpha} e^{\frac{-E(2+\alpha)}{E^{obs}_p}} dE}. $$
(3)

The peak energy flux, denoted by F γ and calculated in erg/cm2/s, is calculated numerically through the following equations:

$$\begin{aligned} F_{\gamma} =& \int_{E_{min}}^{E_{max}}E N(E) dE, \end{aligned}$$
(4)
$$\begin{aligned} =&N_0\int_{E_{min}}^{E_{max}} E\biggl( \frac{E}{E_n}\biggr)^{\alpha}e^{\frac{-E(2+\alpha)}{E^{obs}_p}} dE, \end{aligned}$$
(5)
$$\begin{aligned} =&N_0 K E_n \int_{E_{min}}^{E_{max}} \biggl(\frac{E}{E_n}\biggr)^{\alpha+1} e^{\frac{-E(2+\alpha)}{E^{obs}_p}} dE. \end{aligned}$$
(6)

A factor K=1.6×10−9 is introduced to make the keV-erg conversion and E n =50 keV.

The bolometric luminosity of the 1-second isotropic peak, denoted L iso , is the maximum energy radiated per unit time in all space. It is calculated by integrating the EN E function in the energy band corresponding to the observed gamma radiation band in the source’s frame, i.e.E 1=1 keV to E 2=104 keV. And because of cosmological effects, the corresponding observed energy band is: E 1/(1+z) to E 2/(1+z).

Thus, the k-corrected L iso is calculated via:

$$\begin{aligned} L_{iso} =&4 \pi d_L^2 \int _{E_{1}/(1+z)}^{E_{2}/(1+z)}E N(E)dE, \end{aligned}$$
(7)
$$\begin{aligned} =& 4 \pi d_L^2 F_{\gamma} k_c. \end{aligned}$$
(8)

Here L iso is k-corrected with the method developed by Bloom et al. (2001). Indeed, in Eq. (7) we replace N(E) by Eq. (1) and using Eq. (5) to express N 0, we obtain:

$$\begin{aligned} k_c =&\frac{{\int_{E_{1}/(1+z)}^{E_{2}/(1+z)}(\frac{E}{E_n})^{\alpha+1} e^{\frac{-E(2+\alpha)}{E^{obs}_{p}}} dE}}{{\int_{E_{min}}^{E_{max}}(\frac{E}{E_n})^{\alpha+1} e^{\frac{-E(2+\alpha)}{E^{obs}_{p}}} dE}}, \\ =&\frac{\int_{E_{1}/(1+z)}^{E_{2}/(1+z)}E N(E)dE}{\int_{E_{min}}^{E_{max}}E N(E) dE}, \end{aligned}$$
(9)

where k c being the proper k-correction factor (Yonetoku et al. 2004; Rossi et al. 2008; Elliott et al. 2012).

These integrals are performed numerically using the time-resolved spectral parameters given by Swift. The cosmological distance d L is expressed by the following equation:

$$ d_L=\frac{(1+z)c}{H_0}\int_0^z \frac{dz'}{\sqrt{\varOmega_M(1+z')^3+\varOmega_L}}. $$
(10)

We adopt the following cosmological parameters: Ω M =0.27, Ω L =0.73, and H 0=70 km/s/Mpc (Komatsu et al. 2009).

The total isotropic energy, denoted E iso , which is emitted by a gamma-ray burst in all space, is calculated using the fluences (\(\mathrm{erg/cm^{2}}\)) given by the detectors in the Swift energy band [15–150] keV. To calculate this, we use a cut-off power law spectrum with time-averaged spectral parameters (α m ,E pm ) obtained from the Swift data. Using the CPL function, the fluence S obs , given in the fourth column in Table 4, can be theoretically calculated using the following equation:

$$\begin{aligned} S_{obs} =&\int_{E_{min}}^{E_{max}} E N_i(E) dE, \\ =&N'_0 E_n T_{90}^{obs} \int_{E_{min}}^{E_{max}}\biggl(\frac{E}{E_n} \biggr)^{\alpha_m+1} e^{\frac{-E(2+\alpha_m)}{E^{obs}_{pm}}} dE, \\ =&N'\int_{E_{min}}^{E_{max}}\biggl( \frac{E}{E_n}\biggr)^{\alpha_m+1} e^{\frac{-E(2+\alpha_m)}{E^{obs}_{pm}}} dE , \end{aligned}$$
(11)

with E min =15 keV, E max =150 keV and E n =50 keV. α m and \(E^{obs}_{pm}\) are given in Table 4, for the time-averaged spectrum. N i (E) is the time-integrated spectrum calculated via the product of the time-averaged spectrum by \(T_{90}^{obs}\), the observed duration of the GRB:

$$\begin{aligned} N_i(E) =&T_{90}^{obs}\times \biggl\{ \frac{1}{T_{90}^{obs}}\int_0^{T_{90}^{obs}} N(E,t) dt \biggr\} , \\ =&T_{90}^{obs} \biggl\{ N'_0 \biggl( \frac{E}{E_n}\biggr)^{\alpha_m} e^{\frac{-E(2+\alpha_m)}{E^{obs}_{pm}}} \biggr\} . \end{aligned}$$
(12)

In Eq. (11) the only unknown is the normalization constant N′; it is determined by numerical integration, i.e.:

$$ N' =\frac{S_{obs}}{\int_{E_{min}}^{E_{max}}(\frac{E}{E_n})^{\alpha_m+1} e^{\frac{-E(2+\alpha_m)}{E^{obs}_{pm}}} dE}. $$
(13)

Thus, N′ is used to deduce k-corrected E iso :

$$\begin{aligned} E_{iso} =&\frac{4\pi d_L^2}{1+z} \int_{E_{1}/(1+z)}^{E_{2}/(1+z)}E N_i(E) dE, \\ =& \frac{4 \pi d_L^2}{1+z} S_{obs} k'_c. \end{aligned}$$
(14)

This luminosity is different from the time-resolved peak luminosity that was calculated above.

The (1+z) factor is a cosmological correction that is needed because one must use \(T^{s}_{90}\) (the GRB’s duration in the source’s frame) instead of \(T^{obs}_{90}\): \(T^{s}_{90} = T^{obs}_{90}/(1 + z)\); [E 1=1 keV; E 2=104 keV] is the energy band in the source’s frame. \(k'_{c}\) is the k-correction factor calculated with the parameters of the time-averaged spectrum:

$$\begin{aligned} k'_c =&\frac{{\int_{E_{1}/(1+z)}^{E_{2}/(1+z)}(\frac{E}{E_n})^{\alpha_m+1} e^{\frac{-E(2+\alpha_m)}{E^{obs}_{pm}}} dE}}{{\int_{E_{min}}^{E_{max}}(\frac{E}{E_n})^{\alpha_m+1} e^{\frac{-E(2+\alpha_m)}{E^{obs}_{pm}}} dE}}, \\ =&\frac{\int_{E_{1}/(1+z)}^{E_{2}/(1+z)}E N_i(E)dE}{\int_{E_{min}}^{E_{max}}E N_i(E)dE}. \end{aligned}$$
(15)

4 Results

4.1 Distribution of F γ E iso and L iso

In the previous sections we numerically evaluated the energy flux denoted by F γ  (erg/cm2/s), the peak isotropic bolometric luminosity, L iso , and the isotropic energy E iso . Uncertainties over these quantities are estimated using the Monte Carlo method. We have plotted the distributions P, E iso , L iso in Figs. 2, 3, and 4 respectively. We also obtained the distributions of the two physical quantities E iso and L iso in the sources’ reference frames. In Fig. 4, we note that E iso , which represents the total energy released by the burst during its entire activity, follows a lognormal distribution, previously known (Preece et al. 2000), with a mean equal to 1.7×1052 erg. Most gamma-ray bursts that are detected by other satellites are characterized by this average value.

Fig. 2
figure 2

Histogram of LGRBs in terms of the peak energy flux F γ . We split the interval from log(F γ )=8 to log(F γ )=5 into 8 bins. 57 Swift LGRBs were used in this histogram. Here, we did not consider the constraint on the threshold on the flux: P ph ≥2.6 ph/cm2/s

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Fig. 3
figure 3

Histogram of LGRBs in terms of the peak isotropic luminosity. We split the interval from log(L iso )=49 to 55 into 8 bins. 57 Swift LGRBs were used. Here, we did not consider the constraint: P ph ≥2.6 ph/cm2/s

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Fig. 4
figure 4

Histogram of LGRBs in terms of the isotropic energy. We split the interval from logE iso =49 to 55 into 7 bins. 56 Swift LGRBs were used. Here, we did not consider the constraint: P ph ≥2.6 ph/cm2/s

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In Fig. 5 we present the correlation between L iso and E iso : a burst that releases a large amount of energy is characterized by high luminosity. We note that these two quantities are correlated with a wide dispersion of the observational data.

Fig. 5
figure 5

Plot of logE iso vs. logL iso : 42 Swift LGRBs have been used in this plot (Table 3). Here, we did not consider the constraint: P ph ≥2.6 ph/cm2/s. The error bars over L iso are much larger than those over E iso because of the size of the error bars over the peak flux

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4.2 Correlation relations

One of the most debated issues regarding gamma-ray bursts (GRB ) is the existence of a correlation relation between the spectral parameters of the prompt emission and either the total energy or the luminosity. Three robust correlations have been identified but not yet confirmed. Each involves the peak energy E p of the spectrum νf ν , estimated in the source’s frame. This quantity is strongly correlated with: (a) the total isotropic energy E iso (Amati et al. 2002; Amati and Mon 2006), (b) the isotropic luminosity L iso (Yonetoku et al. 2004), and (c) the total energy collimated with the opening angle θ which is denoted by E θ (Ghirlanda et al. 2004). The opening angle is inferred from the observed break in the temporal profile of the afterglows. In the BATSE observations, θ did not exceed ten degrees, while Swift observations of afterglows do not show, in most cases, such a break. Such correlations apply only in long GRBs. The spectral energy correlations have important implications for both the theoretical understanding of GRBs and for cosmological applications (Ghirlanda et al. 2004, 2005).

4.2.1 The E iso −E p,i relation

The relation between the energies E p,i and E iso , discovered by Amati et al. (2002) has been the subject of several publications, even though it has not yet been fully confirmed. This topic has had several controversies. In Amati et al. (2002) came up with this relationship from a sample of 12 Beppo-SAX GRBs with well-determined redshifts. These researchers showed that there is a purely empirical relation between E p,i , the peak energy of the photon spectrum νf ν of the prompt emission, as measured in the source’s frame and the total equivalent isotropic energy E iso that is emitted in the energy band [1–104 keV], that is the energy radiated by the source in this energy range, assuming an isotropic emission. This relation requires a redshift measurement. The redshift is necessary to know the intrinsic properties of the source, as the intrinsic peak energy is given by the relation:

$$ E_{p,i} = E_p^{obs}\times(1+z), $$
(16)

where \(E^{obs}_{p}\) is the peak energy observed by the Swift/BAT detectors. The Amati relation is given by:

$$ \frac{E_{p,i}}{\mathrm{keV}}=K\times \biggl(\frac{E_{iso}}{10^{52}~\mathrm{erg}} \biggr)^m, $$
(17)

where K and m are constants.

For the original Amati relation, K≈95 and m≈0.5. This relation can be used to constrain cosmological parameters as well as different models aiming to explain the prompt emission. It can also provide information on the nature of the various subclasses of gamma-ray bursts (e.g., LGRB, SGRB, etc.)

We plot our results for the Amati relation in Fig. 6. These plots are for 27 bursts detected by Swift/BAT with well determined redshifts. We plot logE p as a function of logE iso . We represented two extreme lines, in fitting the 27-point distribution. From these two lines we have deduced the mean values for the line’s slope and intercept. In Table 1 we give the constants K and m of Eq. (17), as obtained from the following expressions:

$$\begin{aligned} K =&10^{(52a+b)}, \end{aligned}$$
(18)
$$\begin{aligned} m =&a. \end{aligned}$$
(19)

We also give the original results of Amati et al. (2002).

Fig. 6
figure 6

The E iso −E p,i correlation using 27 Swift LGRBs that satisfy all the constraints that we set

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Table 1 E iso −E p,i correlation
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4.2.2 The L iso −E p,i relation

The relation between the energy E p,i and the isotropic luminosity at peak time, with well-determined redshifts, was found by Yonetoku et al. (2004). It was expressed as:

$$ \frac{L_{iso}}{10^{52}~\mathrm{erg}/\mathrm{s}}=A {\biggl[}\frac{E_{p,i}}{ \mathrm{keV}} { \biggr]}^{p}. $$
(20)

In Fig. 7 we plot E p,i vs. L iso in log–log scale for 27 Swift bursts with well determined redshifts. We have drawn two extreme straight lines which represent “brackets”. From these two lines we deduced the mean values for the slope and the intercept (a and b). In Table 2 we give the constants A and p of Eq. (20), which are obtained from the following relations:

$$\begin{aligned} &{A=10^{-\frac{b}{a}+52},} \end{aligned}$$
(21)
$$\begin{aligned} &{p=a^{-1}.} \end{aligned}$$
(22)

For comparison, we also show the original results of Yonetoku et al. (2004).

Fig. 7
figure 7

The L iso −E p,i correlation using 27 Swift LGRBs that satisfy all the constraints that we set

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Table 2 L iso −E p,i correlation
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4.2.3 Evolution of L iso and E iso with redshift

In the sample of 27 Swift LGRBs, we find an interesting evolution of the isotropic energy E iso in terms of the redshift z. We plot the data in Fig. 8, showing a trend between E iso and z, a trend which can be expressed by the following equation:

$$ \frac{E_{iso}}{\mathrm{erg}}=10^{51.82\pm 0.25} (1+z)^{1.75\pm 0.55}. $$
(23)

This result is in good agreement with the recently published paper (Salvaterra et al. 2013). We also find a similar trend between L iso and z, (Fig. 9), a trend which can be expressed by the following equation:

$$ \frac{L_{iso}}{\mathrm{erg}/\mathrm{s}}=10^{51.2\pm 0.2} (1+z)^{2.6\pm 0.5}. $$
(24)

It is indeed logical to find a (1+z) dependence in the ratio of L iso and E iso due to the cosmological effect on the duration \(T_{90}^{obs} = T_{90}^{s}(1+z)\), which affects only the luminosity.

Fig. 8
figure 8

Evolution of E iso with the redshift z using 27 Swift LGRBs that satisfy all the constraints that we set

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Fig. 9
figure 9

Evolution of L iso with the redshift z using 27 Swift LGRBs that satisfy all the constraints that we set

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5 Discussion

In this section we present a brief review of recent studies that have dealt with the Amati and Yonetoku relations in order to put our study into proper perspective. Some studies (Zhang et al. 2012; Tsutsui et al. 2013) have lately considered whether these correlations apply to both short and long bursts. It had previously been thought that the Amati relation applies only to LGRBs, whereas the Yonetoku relation applies to both. In the study by Zhang and Mészàros (2004) the authors used a sample of 148 LGRBs and 17 SGRBs to investigate this issue for the Yonetoku relation. The results obtained indicate that both the LGRB and SGRB groups seem to adhere to the correlation with the same best-fit: \(L_{iso}\propto E_{p,i}^{1.7}\). This implies that the radiation mechanism is similar for short and long bursts, and probably has a quasi-thermal origin in which most of the energy is dissipated close to the central engine. On the other hand, the study by Tsutsui et al. (2013) considered both the Amati and Yonetoku relations but for short bursts only. The authors first clarified the sometimes ambiguous issue of when a burst is to be considered short. They then distinguished between “secure” and “misguided” SGRBs. Out of an initial sample of 13 bursts, 8 were found to be “secure”. With these 8 bursts they were able to obtain good fits and to show that both the Amati and Yonetoku relations apply; however, for a given E p,i , E iso is dimmer by a factor of about 100, and L iso is dimmer by a factor of about 5 than that known for LGRBs.

Other studies have looked at the possible redshift evolution of these correlations. The study by Geng and Huang (2013) used a sample of 65 bursts to investigate the possible redshift dependence of the low-energy index, α, in the Band function. Their results indicate that such a dependence does exist. Although we did not utilize the Band function in our study, since we used a CPL, our results for the redshift dependence of E iso and L iso are in qualitative agreement with what was found by Geng and Huang (2013).

The study by Nava et al. (2012) used a sample of 47 GRBs to investigate the robustness of the Amati and Yonetoku relations, and also to look into their possible redshift evolution. Although the authors found some outliers, their conclusion was that these relations are genuine and are not due to selection effects. However, they also found no evolution of these correlations with redshift. This final result is in agreement with a recent study (Azzam and Alothman 2013) in which the authors investigate the possible redshift evolution of a sample of 65 bursts by binning the data and carrying out the proper z-correction and k-correction. The authors obtained good fits for the binned data, but found no evidence for redshift evolution.

Our current study is in agreement with the above investigations in that it confirms the existence of the Amati and Yonetoku correlations. However, we have taken a step further by demonstrating that applying a stricter criterion for choosing the GRB sample in the first place, actually improves the quality of these fits, since it reduces the dispersion that is commonly seen in these correlations. Therefore, the proper selection of the data sample is crucial in such studies.

6 Conclusion

We have conducted a statistical study of a sample of Swift bursts. Among the 229 LGRBs with well-determined redshifts, we selected the 71 GRBs whose observed energy \(E^{obs}_{p}\) is within the energy interval 15–150 keV. Among those, 57 GRBs had L iso and E p;i values and could thus be used to test the Yonetoku relation, while 56 GRBs had E iso and E p;i values and could thus be used for the study of the Amati relation. These bursts satisfy constraints on the energy \(E^{obs}_{p}\). The uncertainties (error bars) on the bursts’ physical quantities were estimated using a Monte Carlo method. We present the data for the bursts, along with the error bars, in a summary Table 3. For these bursts, we plotted E iso against E p,i and L iso against E p,i , testing the Amati and Yonetoku relations on that sample. We found the data to be tainted with significant dispersions around the linear trends. But by adding a condition on the peak flux, we obtained a sample of 27 LGRBs for which we got good linearities on those two relations.

Table 3 Flux F γ isotropic energy E iso and luminosity L iso calculated from the Swift data for a sample of 71 GRBs that satisfy our constraints, except for the threshold on the photon flux
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Table 4 Data for 71 Swift GRBs
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