New measurements of Ω_{m} from gammaray bursts
^{1}
Dip. di Fisica, Sapienza Universitá di Roma,
Piazzale Aldo Moro 5,
00185
Rome,
Italy
^{2}
ICRANetPescara, Piazza della Repubblica 10,
65122
Pescara,
Italy
email: luca.izzo@icra.it; muccino@icra.it
^{3}
ICRANetRio, Centro Brasileiro de Pesquisas Fisicas, Rua Dr.
Xavier Sigaud 150, 22290180
Rio de Janeiro,
Brazil
email: elena.zaninoni@gmail.com
^{4}
INAF–Istituto di Astrofisica Spaziale e Fisica Cosmica, Bologna,
via Gobetti 101, 40129
Bologna,
Italy
^{5}
INAF–Napoli, Osservatorio Astronomico di Capodimonte, Salita
Moiariello 16, 80131
Napoli,
Italy
Received:
4
May
2015
Accepted:
24
August
2015
Context. Data from cosmic microwave background radiation (CMB), baryon acoustic oscillations (BAO), and supernovae Ia (SNeIa) support a constant dark energy equation of state with w_{0} ~ −1. Measuring the evolution of w along the redshift is one of the most demanding challenges for observational cosmology.
Aims. We discuss the existence of a close relation for gammaray bursts (GRBs), named Comborelation, based on characteristic parameters of GRB phenomenology such as the prompt intrinsic peak energy E_{p,i}, the Xray afterglow initial luminosity L_{0} and the restframe duration τ of the shallow phase, and the index of the late powerlaw decay α_{X}. We use it to measure Ω_{m} and the evolution of the dark energy equation of state. We also propose a new calibration method for the same relation, which reduces the dependence on SNe Ia systematics.
Methods. We have selected a sample of GRBs with 1) a measured redshift z; 2) a determined intrinsic prompt peak energy E_{p,i}; and 3) a good coverage of the observed (0.3–10) keV afterglow light curves. The fitting technique of the restframe (0.3–10) keV luminosity light curves represents the core of the Comborelation. We separate the early steep decay, considered a part of the prompt emission, from the Xray afterglow additional component. Data with the largest positive residual, identified as flares, are automatically eliminated until the pvalue of the fit becomes greater than 0.3.
Results. We strongly minimize the dependency of the ComboGRB calibration on SNe Ia. We also measure a small extraPoissonian scatter of the Comborelation, which allows us to infer from GRBs alone Ω_{M} = 0.29^{+0.23}_{0.15} (1σ) for the ΛCDM cosmological model, and Ω_{M} = 0.40^{+0.22}_{0.16}, w_{0} = −1.43^{+0.78}_{0.66} for the flatUniverse variable equation of state case.
Conclusions. In view of the increasing size of the GRB database, thanks to future missions, the Comborelation is a promising tool for measuring Ω_{m} with an accuracy comparable to that exhibited by SNe Ia, and to investigate the dark energy contribution and evolution up to z ~ 10.
Key words: cosmological parameters / gammaray burst: general / dark energy
© ESO, 2015
1. Introduction
Gammaray bursts (GRBs) are observed in a wide range of spectroscopic and photometric redshifts, up to z ~ 9 (Salvaterra et al. 2009; Tanvir et al. 2009; Cucchiara et al. 2011). This suggests that GRBs can be used to probe the highz Universe, in terms of investigating the reionization era, population III stars, the metallicity of the circumburst medium, the faintend of galaxies luminosity evolution (D’Elia et al. 2007; Robertson & Ellis 2012; Macpherson et al. 2013; Trenti et al. 2013, 2015), and as cosmological rulers (e.g. Ghirlanda et al. 2004; Dai et al. 2004; Amati & Della Valle 2013). The latter works show that GRBs, through the correlation between radiated energy or luminosity and the photon energy at which their νF(ν) spectrum peaks E_{p,i}, admittedly with a lower level of accuracy, provide results in agreement with supernovae Ia (SN Ia, Perlmutter et al. 1998, 1999; Schmidt et al. 1998; Riess et al. 1998), baryonic acoustic oscillations (BAO, Blake et al. 2011), and cosmic microwave background (CMB) radiation (Planck Collaboration XVI 2014). The Universe is spatially flat (e.g. de Bernardis et al. 2000), and it is dominated by a still unknown vacuum energy, usually called dark energy, which is responsible for the observed acceleration. Measuring the equation of state (EOS, ω = p/ρ, with p the pressure and ρ the density of the dark energy) is one of the most difficult tasks in observational cosmology today. Current data (Suzuki et al. 2012; Planck Collaboration XIII 2015) suggest that w_{0} ~ −1 and w_{a} ~ 0, the expected values for the cosmological constant. Although these results are probably sufficient to exclude a very rapid evolution of dark energy with z, we cannot yet exclude that it may evolve with time, as originally proposed by Bronstein (1933). In principle, with SNe Ia we can push our investigation up to z ≈ 1.7 (Suzuki et al. 2012). However we note that the possibility of detecting SNe Ia at higher redshifts will depend on the availability of nextgeneration telescopes and also on the time delay distribution of SNe Ia (see Fig. 8 in Mannucci et al. 2006).
A solution to this problem may be provided by GRBs: their redshift distribution peaks around z ~ 2−2.5 (Coward et al. 2013) and extends up to the photometric redshift of z = 9.4, (Cucchiara et al. 2011). Therefore, given this broad range of z and their very high luminosities, GRBs are a class of objects suitable to explore the trend of dark energy density with time (Lloyd & Petrosian 1999; RamirezRuiz & Fenimore 2000; Reichart et al. 2001; Norris et al. 2000; Amati et al. 2002, 2008; Ghirlanda et al. 2004; Dai et al. 2004; Yonetoku et al. 2004; Firmani et al. 2006; Liang & Zhang 2006; Schaefer 2007; Capozziello & Izzo 2008; Dainotti et al. 2008; Tsutsui et al. 2009; Wei et al. 2014; Wang et al. 2015). There are two complications connected with these approaches. First, the correlations are always calibrated by using the entire range of SNe Ia up to z = 1.7; therefore, this procedure strongly biases the GRB cosmology and for our purposes we need the highest level of independent calibration possible. Second, the data scatter of these correlations is not tight enough to constrain cosmological parameters (in this work we refer to the different extrascatters published in Margutti et al. 2013), even when a large GRB dataset is used. In this work we present a method that can override the latter and minimize the former.
Fig. 1 E_{X,iso}–E_{γ,iso}–E_{p,i} correlation proposed by Bernardini et al. (2012) and Margutti et al. (2013) (courtesy R. Margutti). 

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Very recently Bernardini et al. (2012, hereafter B12) and Margutti et al. (2013, hereafter M13) have published an interesting correlation that connects the prompt and the afterglow emission of GRBs (see Fig. 1). This relation strongly links the Xray and γray isotropic energy with the intrinsic peak energy, , unveiling an interesting connection between the early, more energetic component of GRBs (E_{γ,iso} and E_{p,i}) and their late emission (E_{X,iso}), opportunely filtered for flaring activity. In addition, the intrinsic extra scatter is very small σ_{EX,iso} = 0.30 ± 0.06 (1σ). Starting from the results obtained by B12 and M13, and combining them with the wellknown E_{p,i} − E_{iso} relation (Amati et al. 2002), we present here a new correlation involving four GRB parameters, which we named the Comborelation. This new relation is then used to measure Ω_{m} and to explore the possible evolution of w of EOS with the redshift by using as a “candle” the initial luminosity L_{0} of the shallow phase of the afterglow, which we find to be strictly correlated with quantities directly inferred from observations.
The paper is organized as follows. In Sect. 2 we describe in detail how we obtain the formulation of the newly proposed GRB correlation, and present the sample of GRBs used to test our results, the procedure for the fitting of the Xray afterglow light curves, and finally, the existence of the correlation assuming a standard cosmological scenario. Then, in Sect. 3, we discuss the use of the relation as cosmological parameter estimator, which involves a calibration technique that does not require the use of the entire sample of SNe Ia, as was often done in previous similar works. In Sect. 4, we test the use of GRBs estimating the main cosmological parameters, as well as the evolution of the dark energy EOS. Finally, we discuss the final results in the last section.
2. The Comborelation
We present here a new GRB correlation, the Comborelation, obtained after combining E_{γ,iso}–E_{X,iso}–E_{p,i} (B12 and M13), the E_{γ,iso}–E_{p,i} (Amati et al. 2002) correlations, and the analytical formulation of the Xray afterglow component given in Ruffini et al. (2014, hereafter R14).
The threeparameter scaling law reported in B12 and M13 can be generally written as (1)where E_{X,iso} is the isotropic energy of a GRB afterglow in the restframe (0.3–30) keV energy range obtained by integrating the light curve in luminosity over a specified time interval, E_{γ,iso} is the isotropic energy of a GRB prompt emission, and E_{p,i} is the intrinsic spectral peak energy of a GRB. Since E_{X,iso} and E_{γ,iso} are both cosmologicaldependent quantities, we reformulate this correlation in order to involve only one cosmologicaldependent quantity.
The right term of Eq. (1) can be rewritten using the wellknown formulation of the Amati relation, , which provides (2)where γ = η × β − δ and A is the normalization constant. Since the Amati relation is valid only for long bursts, in the following discussion we will exclude short GRBs with a restframe T_{90} duration smaller than 2 s, and short bursts with “extended emission” (Norris & Bonnell 2006), or “disguised short” GRBs (Bernardini et al. 2007; Caito et al. 2010), which have hybrid characteristic between short and long bursts.
To rewrite Eq. (2) for cosmological purposes, we have calculated the total isotropic Xray energy in the restframe (0.3–10) keV energy range by integrating the Xray luminosity , expressed as a function of the cosmological restframe arrival time , over the time interval . This luminosity is obtained by considering four steps (see e.g. Appendix A and Pisani et al. 2013).
It is well known that the Xray afterglow phenomenology can be described by the presence of an additional component emerging from the soft Xray steep decay of the GRB prompt emission, and characterized by a first shallow emission, usually named the plateau, and a late powerlaw decay behaviour (Nousek et al. 2006; Zhang et al. 2006; Willingale et al. 2007). In addition, many GRB Xray light curves are characterized by the presence of large, latetime flares, whose origin is very likely associated with latetime activity of the internal engine (Margutti et al. 2010). Since their luminosities are much lower than the prompt one, we exclude Xray flares from our analysis via a lightcurve fitting algorithm, which will be explained later in the text. We then make the assumption that the E_{X,iso} quantity refers only to the component whose Xray luminosity is given by the phenomenological function defined in R14 (3)where L_{0}, τ, and α_{X} are, respectively, the luminosity at , the characteristic timescale of the end of the shallow phase, and the late powerlaw decay index of a GRB afterglow.
Therefore, if we extend the integration time interval to and , the integral of the function in Eq. (3) gives (4)with the requirement that α_{X}< −1. This condition is necessary to exclude divergent values of E_{X,iso} computed from Eq. (4), for . It is worth noting that light curves providing values α_{X}> −1 could have a change in slope at very late times (beyond the XRT time coverage) and/or, in principle, could be polluted by a late flaring activity, resulting in a less steep late decay.
Considering Eqs. (2) and (4), we can finally formulate the following relation between GRB observables (5)which we name the Comborelation. At first sight, Eq. (5) suggests the existence of a physical connection between specific physical properties of the afterglow and prompt emission in long GRBs. A twodimensional fashion of the correlation in Eq. (5) in logarithmic units can be written as (6)where the set of parameters in the square brackets has the meaning of a logarithmic independent coordinate, and in the following will be expressed as log ^{[}X(γ,E_{p,i},τ,α_{X})^{]}.
We have tested the reliability of Eqs. (5) and (6) by building a sample of GRBs satisfying the following restrictions:

a measured redshift z;

a determined prompt emission spectral peak energy E_{p,i};

a complete monitoring of the GRB Xray afterglow light curve from the early decay ( s, when present) until late emission (–10^{6} s).
We start the analysis by computing the restframe (0.3–10) keV energy band light curves (see Appendix A). The fitting of the continuum part of the Xray light curves was performed by using a semiautomated procedure based on the χ^{2} statistic, which eliminates the flaring part (see e.g. Appendix B and Margutti et al. 2011; Zaninoni 2013, and M13, for details). A total of 60 GRBs are found, whose distribution in the Comborelation plane is shown in Fig. 2, where the value of the luminosity L_{0} for each GRB is calculated from the flat ΛCDM scenario. The corresponding bestfit parameters, as well as the extrascatter term σ_{ext}, have been derived by following the procedure by D’Agostini (2005), and are respectively log [A/ (erg / s)] = 49.94 ± 0.27, γ = 0.74 ± 0.10, and σ_{ext} = 0.33 ± 0.04. The Spearman’s rank correlation coefficient is ρ_{S} = 0.92, while the pvalue computed from the twosided Student’s tdistribution, is p_{val} = 9.13 × 10^{22}.
Fig. 2 Correlation considering the entire sample of 60 GRBs. The green empty boxes are the data of each of the sources, derived as described in Appendix B, the solid black line is the best fit of the data, while the dotted grey lines and the dashed grey lines correspond, respectively, to the dispersion on the correlation at 1σ_{ex} and 3σ_{ex}. 

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3. Calibration of the Comborelation
The lack of very nearby (z ~ 0.01) GRBs prevents the possibility of calibrating GRB correlations, as is usually done with SNe Ia. In recent years different methods have been proposed to avoid this “circularity problem” in calibrating GRB relations (Kodama et al. 2008; Demianski et al. 2012, and references therein). The common approach uses an interpolating function for the distribution of luminosity distances (distance moduli) of SNe Ia with redshift, and then extending it to GRBs at higher redshifts. In the following we introduce and describe an alternative method for calibrating the Comborelation (see also Ghirlanda et al. 2006) which consists of two steps: 1) we identify a small but sufficient subsample of GRBs that lie at the same redshift, and then infer the slope parameter γ from a bestfit procedure; 2) once we determine γ, the luminosity parameter A can be obtained from a direct comparison between the nearest, z = 0.145, GRBs in our sample and SNe Ia located at very similar redshifts. We note that this approach is different from previous ones in that we do not use the whole redshift range covered by SNe Ia, but we limit our calibration analysis to z = 0.145 where the effect of the cosmology on the distance modulus of the calibrating SNe Ia is small (see Fig. 3).
Fig. 3 Residual distance modulus μ_{obs} − μ_{th} for different values of the density cosmological parameters Ω_{m} and Ω_{Λ} up to z = 2.0. We consider the best fit to be the standard ΛCDM model, where Ω_{m} = 0.27, Ω_{Λ} = 0.73, and H_{0} = 71 km s^{1}/Mpc (black line). Union2 SNe Ia data residuals are shown in grey. The large spread (more than 1 mag) shown by μ at z = 1.5 and at z = 0.145 (the two vertical dashed lines) where the scatter is almost 0.2 mag is clearly evident. 

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3.1. The determination of the slope γ
Five GRBs used for the determination of the slope parameter γ of the correlation.
The existence of a subsample with a sufficient number of GRBs lying at almost the same cosmological distance would, in principle, allow us to infer γ, overriding any possible cosmological dependence (assuming a homogeneous and isotropic Universe). In our sample of 60 GRBs there is a small subsample of 5 GRBs located at a very similar redshift, see Table 1. The difference between the maximum redshift of this 5GRB sample (z = 0.544) and the minimum one (z = 0.5295) corresponds to a variation of 0.015 in redshift and of 0.07 in distance modulus μ in the case of the standard ΛCDM model. This very small difference is sufficient for our purposes.
To avoid any possible cosmological contamination, we do not consider the luminosity L_{0} as the dependent variable, but we consider instead the energy flux F_{0}, which is related to the luminosity through the expression . This assumption does not influence the final result, since d_{l} is almost the same for the 5GRB sample and, therefore, the term can be absorbed in the normalization constant. Consequently, we build the energy flux light curve for each GRB, and then, following the same procedure described in Sect. 2, we perform a bestfit analysis of this subsample of five GRBs using the maximum likelihood technique. From the best fit we find a value of γ = 0.89 ± 0.15, with an extra scatter of σ_{ext} = 0.40 ± 0.04, see Fig. 4.
Fig. 4 Correlation found for the sample of five GRBs located at the same redshift. The blue triangles are the data of each of the five sources, derived from the procedure in Appendix B. The solid line represents the best fit while the dashed line is the dispersion on the correlation at 1σ_{ex}. 

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3.2. Calibration of A with SNe Ia
Among the considered 60 GRBs, the nearest one is GRB 130702A at z = 0.145. We use it to perform the calibration of the Comborelation with SNe Ia located at the same distance. It is clear that at these redshifts (z = 0.145) the offset provided by SNe Ia is much smaller than the value inferred from SNe Ia at larger redshifts, see Fig. 3. In this light we have selected Union2 SNe Ia (Amanullah et al. 2010; Suzuki et al. 2012) with redshift between z = 0.143 and z = 0.147. We find five SNe Ia that satisfy this condition; their properties are shown in Table 2. We then compute the average value of their distance modulus and its uncertainty, ⟨ μ ⟩ = 39.19 ± 0.27, which will be used for the final calibration. The parameter A can be obtained inverting Eq. (6), and considering L_{0} = 4πdl^{2}F_{0}: (7)The generic SN distance modulus μ can be directly related to the luminosity distance d_{l} by
Five SNe Ia selected from the Union2 sample (Amanullah et al. 2010; Suzuki et al. 2012) and used for the calibration of the GRB correlation.
(8)where the last equality takes into account the fact that d_{l} is expressed in cm. Substituting this last expression for in Eq. (7) we obtain (9)where the term ψ(γ,E_{p.i},τ,α_{X},F_{0}) comprises the last three terms on the right hand side of Eq. (7). Substituting the quantities for GRB 130702A in Eq. (9), and the value of ⟨ μ ⟩ previously obtained, we infer a value for the parameter log [ A/ (erg / s) ] = 49.54 ± 0.20, where the uncertainty also takes the σ_{ext} value found above into account.
4. Cosmology with the Comborelation
4.1. Building the GRB Hubble diagram
We now discuss the possible use of the proposed GRB Comborelation to measure the cosmological constant and the mass density, as well as their evolution with redshift z. The possibility of estimating the luminosity distance d_{l} from the GRB observable quantities allows to us define a distance modulus for GRBs, and then its uncertainty, as (10)where . The quantity μ_{GRB} can be directly compared with the theoretical cosmological expected value μ_{th}, which depends on the density parameters Ω_{m} and Ω_{Λ}, the curvature term Ω_{k} = 1 − Ω_{m} − Ω_{Λ}, and the Hubble constant H_{0}(11)The luminosity distance d_{l} is defined as (12)with E(z,Ω_{m},Ω_{Λ},w(z)) = Ω_{m}(1 + z)^{3} + Ω_{Λ}(1 + z)^{3(1 + w(z))} + Ω_{k}(1 + z)^{2} (see e.g. Goobar & Perlmutter 1995). In the following we fix the Hubble constant at the recent value inferred from lowredshift SNe Ia, corrected for star formation bias, and calibrated with the LMC distance (Rigault et al. 2014): H_{0} = 70.6 ± 2.6.
The corresponding uncertainties on μ_{GRB} were computed considering an observed term, σμ_{obs}, which takes into account each uncertainty on the observed quantities of the Comborelation, e.g. F_{0},τ,α, and E_{p,i}, and a “statistical” term, σμ_{rel}, which takes into account the uncertainties on the parameters of the Comborelation, A and γ, and the weight of the extra scatter value σ_{ext}. The final uncertainty on each single GRB distance modulus (13)allows us to build the ComboGRB Hubble diagram (see also Izzo et al. 2009), which is shown in Fig. 5. It is possible to quantify the reliability of any cosmological model with our sample of 60 GRBs, which represents a unique dataset from low redshift (z = 0.145) to very large distances (z = 8.23). To reach our goal, we make the fundamental assumption that our GRB sample is normally distributed around the bestfit cosmology, which we are going to estimate. With this hypothesis, we consider as teststatistic the chisquare test, which is defined as (14)where , μ_{th}(z,Ω_{m},Ω_{Λ},w(z)), and σμ_{GRB} are respectively defined in Eqs. (10), (11), and (13). The cosmology is included in the quantity μ_{th}(z,Ω_{m},Ω_{Λ},H_{0}), which we allow to vary. To determine the best configuration of parameters that most closely fits the distribution of GRBs in the Hubble diagram we maximize the loglikelihood function, −2ln(e^{χ2}), which is equivalent to the minimization of the function defined in Eq. (14).
Fig. 5 ComboGRB Hubble diagram. The black line represents the best fit for the function μ(z) as obtained by using only GRBs and for the case of the ΛCDM scenario. 

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4.2. Fit results
4.2.1. ΛCDM case
In the ΛCDMmodel, the energy function E(z,Ω_{m},Ω_{Λ},w(z)) is characterized by an EOS for the dark energy term fixed to w = −1. Since we have that Ω_{m} + Ω_{Λ} + Ω_{k} = 1, we vary the matter and cosmological constant density parameters, also obtaining in this way an estimate of the curvature term. We obtain that GRBs alone provide , see also Fig. 6.
Fig. 6 1σ(Δχ^{2} = 2.3) confidence region in the (Ω_{m}, Ω_{Λ}) plane for the ComboGRB sample (dark blue), and for the total (60 observed + 300 MC simulated GRBs, (light blue). 

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4.2.2. Variable w_{0} case
In a flat Universe (Ω = Ω_{m} + Ω_{Λ} = 1) with a constant value of w_{0} different from the standard value w = −1, we can provide useful constraints for alternative dark energy theories. In this case, we only vary the matter density and the dark energy equations of state, obtaining an estimate of the density matter of and of a dark energy EOS parameter .
4.2.3. Evolution of w(z)
An interesting casestudy consists of a timeevolving dark energy EOS in a flat cosmology, since the evolution of w(z) can be directly studied with GRBs at larger redshifts. We consider an analytical formulation for the evolution of w with the redshift, which was proposed by Chevallier & Polarski (2001) and Linder (2003) (CPL), and where the w(z) can be parameterized by (15)The CPL parameterization implies that for large z the w(z) term tends to the asymptotic value w_{0} + w_{a}. Using a sample extending at large redshifts, e.g. GRBs, will allow a better estimate of these parameters since the effects of a varying w(z) on the distance modulus are more evident at redshift z ≥ 1. The GRB data provide a bestfit result .
4.2.4. Perspective: a Monte Carlo simulated sample of GRBs
The current sample of GRBs that satisfies the Comborelation is quite limited (60 bursts), when compared to the sample in Amati & Della Valle (2013) (~200 bursts) or the SNe sample in the Union 2.1 release (Suzuki et al. 2012). A more numerous sample can help to understand whether the Comborelation can provide better constraints on the cosmological parameters. To this aim, following the prescription of Li (2007), we used Monte Carlo (MC) simulations to generate a sample of 300 synthetic GRBs satisfying the Comborelation. This value comes from the expected number of GRBs detected in five years of operations of current (Swift) and future (SVOM, Gotz et al. 2009, and LOFT, Feroci et al. 2012) missions dedicated to observing GRBs.
First, we fitted the lognormal distributions of the 60 observed z, E_{p,i}, τ, and  α + 1 , (16)where ξ = z, E_{p,i}, τ, and  α + 1 , and we found the following mean values and dispersions: μ_{z} ± σ_{z} = 0.26 ± 0.27, μ_{Ep,i} ± σ_{Ep,i} = 2.54 ± 0.40, μ_{τ} ± σ_{τ} = 2.85 ± 0.65, and μ_{ α + 1 } ± σ_{ α + 1 } = −0.36 ± 0.31. Then, from these distributions we computed log X(γ,E_{p.i},τ,α) and we generated the initial luminosity log L_{0} = γlog X + A from the frequency distribution (17)assuming that the Comborelation is independent of the redshift and considering its extrascatter σ_{ext}. The values of γ, A, and σ_{ext} are reported in Sect. 3. Finally, to complete the set of parameters necessary to compute the distance modulus of the simulated sample of GRBs from each pair (log L_{0}, z), we generated the corresponding log F_{0} (μ_{F0} ± σ_{F0} = −9.87 ± 0.85). In the following, the attached errors on the MC simulated Comborelation parameters will be taken as 30% of their corresponding values, which reflects the uncertainty of the “real” GRB sample.
We have verified whether the constraints on (Ω_{m}, Ω_{Λ}) will improve by using a larger sample of 360 GRBs (the real sample of 60 GRBs observed and a MCsimulated sample of 300 GRBs, described above). The improvement on the constraints on Ω_{m} and Ω_{Λ} is clear: the uncertainties on the density parameters improve considerably ( for the ΛCDM case, for the variable w_{0} case), as is also clear in the contour plots shown in Figs. 6 and 7.
Fig. 7 1σ(Δχ^{2} = 2.3) confidence region in the (Ω_{m}, w_{0}) plane for the ComboGRB sample (dark blue), and for the total of 60 observed + 300 MC simulated GRBs (light blue). The black dashed line represents the 1σ confidence region obtained using the recent Union 2.1 SNe Ia sample (Suzuki et al. 2012). 

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4.3. GRBs compared with SNe, CMBs, BAOs
In order to compare the ComboGRB sample results, we also consider the following datasets:

The measurements of the baryon acoustic peaks A_{obs} = (0.474 ± 0.034,0.442 ± 0.020,0.424 ± 0.021) at the corresponding redshifts z_{BAO} = (0.44,0.6,0.73) in the galaxy correlation function as obtained by the WiggleZ dark energy Survey (Blake et al. 2011). The BAO peak is defined as (Eisenstein et al. 2005) (18)where r(z,Ω_{m},Ω_{Λ},w(z)) is the comoving distance. The bestfit cosmological model is determined by the minimization of the corresponding chisquared quantity (19)where C^{1} is the inverse covariance matrix of the measurements of the WiggleZ survey (Blake et al. 2011).

The measurement of the shift parameter R_{obs} = 1.7407 ± 0.0094 as obtained from the Planck first data release (Planck Collaboration XVI 2014). The R quantity is the least cosmological modeldependent parameter (particularly from H_{0}) that can be extracted from the analysis of the CMB (Wang & Mukherjee 2006) and is defined as (20)where z_{rec} is the redshift of the recombination. The bestfit cosmological model is determined by the minimization of the corresponding chisquared term (21)
Fig. 8 1σ(Δχ^{2} = 2.3) confidence region in the (Ω_{m}, Ω_{Λ}) plane for the observed GRB sample (blue), with the inclusion of the MCsimulated 300 GRBs sample (cyan), and with the samples of SNe (grey), BAOs (red), and CMBs (green). The dashed line represents the condition of the Flat Universe Ω_{m} + Ω_{Λ} = 1. 

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5. Discussions and conclusions
In this paper we have presented the “Combo” relation, a new tool for GRB cosmology. This relationship provides a very close link between prompt and afterglow parameters and it is characterized by a small intrinsic scatter, which makes this correlation very suitable for cosmological purposes. We recognize the fundamental role of the SwiftXRT (Gehrels et al. 2004; Burrows et al. 2005) which, thanks to its ability to slew very rapidly toward the location of a GRB event, provides realtime and detailed data of GRB afterglow light curves, whose evolution is at the base of the proposed Comborelation. From our analysis the following results emerge:

The proposed twostep calibration of the ComboGRB relation greatly minimizes the dependence on SNe Ia.

GRBs data alone provide for the ΛCDM case , see Fig. 8.

A recent paper (Milne et al. 2015) highlights the existence of an observational bias (a systematic difference in the velocity of SNe Ia ejecta, which is reflected in their curves), potentially affecting the measurements of cosmological parameters obtained with SNe Ia. On the basis of our results we conclude that given the current accuracy of GRB measurements we cannot exclude, within the errors, that an effect like this is at play; however, this effect should not change the conclusions derived from SNeIa observations.

The launch of advanced and more sensitive detectors, such as the incoming SVOM (Gotz et al. 2009) and the proposed LOFT (Feroci et al. 2012) missions (and the expected Swift operations in the near future), will dramatically increase the number of GRBs in the dataset. In five years of operation of the SVOM mission alone, we expect to reach a “good enough” sample of 300 GRBs. With a Monte Carlo simulated sample of 300 GRBs, we will significantly improve the accuracy of Ω_{m} measurement, up to , which is comparable with type Ia SNe (Suzuki et al. 2012).

By using the CPL analytical parameterization, adopted to study the evolution of the dark energy EOS (see Eq. (15)), we find , , and .

The analysis of a combined (SNe+BAO+CMB) dataset confirms that the increasing size of the GRB sample will improve the accuracy of the measurement of the Ω_{m} parameter and in particular of the evolution of w up to z ~ 10.

The analytical expression of the Comborelation provides an explicit close link, characterized by a small intrinsic scatter, between the prompt and the afterglow GRB emissions. This points out the existence of a physical connection between the prompt and the afterglow emissions, which represents a new challenge for GRB models.
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Acknowledgments
We thank the referee for her/his constructive comments which have improved the paper. E.Z. acknowledges the support by the International Cooperation Program CAPESICRANet financed by CAPES – Brazilian Federal Agency for Support and Evaluation of Graduate Education within the Ministry of Education of Brazil. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.
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Appendix A: Computation of the restframe 0.3–10 keV luminosity L(t)
The restframe 0.3–10 keV luminosity was obtained by considering four steps.

(1)
We obtained the SwiftXRT flux light curves in the observed 0.3–10 keV energy band^{2}.

(2)
We transformed the observed flux f_{obs} from the observed energy band 0.3–10 keV to the restframe energy band 0.3–10 keV by assuming an absorbed powerlaw function as the best fit for the spectral energy distribution of the XRT data, N(E) ~ E^{− γ}, with Galactic and intrinsic column densities obtained from the H i radio map (Kalberla et al. 2005) and from the best fit of the total afterglow spectrum, respectively. By using the photon indexes inferred for each time interval, the restframe flux light curve f_{rf} is given by (A.1)

(3)
We transformed the observed time t_{a} into the restframe time by correcting for z(A.2)
 (4)
Appendix B: Determination of the sample and verification of the correlation
To obtain the parameters involved in the Eq. (5), we needed to select an adequate sample of GRBs, to fit their Xray light curves, and to collect or to calculate their E_{p,i} values. The selection criteria have already been delineated in Sect. 2.
The entire procedure works in the restframe 0.3–10 keV energy range for all GRBs. For the Xray light curve fitting technique we developed a semiautomated code performing all the needed operations. The code is based on the IDL^{3} language, and the fitting routine used is MPFIT^{4} (Markwardt 2009), which is based on the nonlinear least squares fitting. First, the procedure fits the complete light curve, then it eliminates at every iteration the data point with the largest positive residual, until it obtains a fit with a pvalue greater than 0.3. To fit the light curves considered in luminosity units (erg/s), we use the composite function (R14):

1.
a power law for the early steep decay (B.1)with L_{p} the normalization factor and α_{p} the slope;

2.
Eq. (3) for the afterglow additional component.
An application of this joint fitting procedure is shown in Fig. B.1 for the case of GRB 060418. After the fitting procedure, we select only the GRBs with α_{X}< −1, a condition necessary for the convergence of the integral in Eq. (4). The final total sample, summarized in Table B.1, consists of 60 GRBs.
Fig. B.1 Example of the combined fitting procedure (solid red line) as described in Eqs. (3) and (B.1), filtered by the flares. The early steep decay fitted by using the powerlaw function in Eq. (B.1) is indicated by the purple dashed line, while the afterglow additional component is fitted by the phenomenological function in Eq. (3) (see also R14), and described by the dotdashed cyan curve. In this specific case, the luminosity light curve of GRB 060418 is shown in which the black dots with the error bars are the flarefree data, the grey dots are the excluded data recognized as due to the flares. The vertical green dotted line indicates the characteristic timescale of the parameter τ. 

Open with DEXTER 
Long bursts with α_{X}< −1 analysed in this work (first column) and their main parameters.
All Tables
Five GRBs used for the determination of the slope parameter γ of the correlation.
Five SNe Ia selected from the Union2 sample (Amanullah et al. 2010; Suzuki et al. 2012) and used for the calibration of the GRB correlation.
Long bursts with α_{X}< −1 analysed in this work (first column) and their main parameters.
All Figures
Fig. 1 E_{X,iso}–E_{γ,iso}–E_{p,i} correlation proposed by Bernardini et al. (2012) and Margutti et al. (2013) (courtesy R. Margutti). 

Open with DEXTER  
In the text 
Fig. 2 Correlation considering the entire sample of 60 GRBs. The green empty boxes are the data of each of the sources, derived as described in Appendix B, the solid black line is the best fit of the data, while the dotted grey lines and the dashed grey lines correspond, respectively, to the dispersion on the correlation at 1σ_{ex} and 3σ_{ex}. 

Open with DEXTER  
In the text 
Fig. 3 Residual distance modulus μ_{obs} − μ_{th} for different values of the density cosmological parameters Ω_{m} and Ω_{Λ} up to z = 2.0. We consider the best fit to be the standard ΛCDM model, where Ω_{m} = 0.27, Ω_{Λ} = 0.73, and H_{0} = 71 km s^{1}/Mpc (black line). Union2 SNe Ia data residuals are shown in grey. The large spread (more than 1 mag) shown by μ at z = 1.5 and at z = 0.145 (the two vertical dashed lines) where the scatter is almost 0.2 mag is clearly evident. 

Open with DEXTER  
In the text 
Fig. 4 Correlation found for the sample of five GRBs located at the same redshift. The blue triangles are the data of each of the five sources, derived from the procedure in Appendix B. The solid line represents the best fit while the dashed line is the dispersion on the correlation at 1σ_{ex}. 

Open with DEXTER  
In the text 
Fig. 5 ComboGRB Hubble diagram. The black line represents the best fit for the function μ(z) as obtained by using only GRBs and for the case of the ΛCDM scenario. 

Open with DEXTER  
In the text 
Fig. 6 1σ(Δχ^{2} = 2.3) confidence region in the (Ω_{m}, Ω_{Λ}) plane for the ComboGRB sample (dark blue), and for the total (60 observed + 300 MC simulated GRBs, (light blue). 

Open with DEXTER  
In the text 
Fig. 7 1σ(Δχ^{2} = 2.3) confidence region in the (Ω_{m}, w_{0}) plane for the ComboGRB sample (dark blue), and for the total of 60 observed + 300 MC simulated GRBs (light blue). The black dashed line represents the 1σ confidence region obtained using the recent Union 2.1 SNe Ia sample (Suzuki et al. 2012). 

Open with DEXTER  
In the text 
Fig. 8 1σ(Δχ^{2} = 2.3) confidence region in the (Ω_{m}, Ω_{Λ}) plane for the observed GRB sample (blue), with the inclusion of the MCsimulated 300 GRBs sample (cyan), and with the samples of SNe (grey), BAOs (red), and CMBs (green). The dashed line represents the condition of the Flat Universe Ω_{m} + Ω_{Λ} = 1. 

Open with DEXTER  
In the text 
Fig. B.1 Example of the combined fitting procedure (solid red line) as described in Eqs. (3) and (B.1), filtered by the flares. The early steep decay fitted by using the powerlaw function in Eq. (B.1) is indicated by the purple dashed line, while the afterglow additional component is fitted by the phenomenological function in Eq. (3) (see also R14), and described by the dotdashed cyan curve. In this specific case, the luminosity light curve of GRB 060418 is shown in which the black dots with the error bars are the flarefree data, the grey dots are the excluded data recognized as due to the flares. The vertical green dotted line indicates the characteristic timescale of the parameter τ. 

Open with DEXTER  
In the text 