New measurements of Ω m from gamma-ray bursts
Luca Izzo, Marco Muccino, Elena Zaninoni, Lorenzo Amati, Massimo Della Valle
AAstronomy & Astrophysics manuscript no. Combo˙LEt c (cid:13)
ESO 2018September 18, 2018
New measurements of Ω m from gamma–ray bursts L. Izzo , , M. Muccino , , E. Zaninoni , L. Amati , M. Della Valle , Dip. di Fisica, Sapienza Universit´a di Roma, Piazzale Aldo Moro 5, I-00185 Rome, Italy. ICRANet-Pescara, Piazza della Repubblica 10, I-65122 Pescara, Italy. ICRANet-Rio, Centro Brasileiro de Pesquisas Fisicas, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil. INAF, Istituto di Astrofisica Spaziale e Fisica Cosmica, Bologna, Via Gobetti 101, I-40129 Bologna, Italy. INAF-Napoli, Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, I-80131 Napoli, Italy.
ABSTRACT
Context.
Data from cosmic microwave background radiation (CMB), baryon acoustic oscillations (BAO), and supernovae Ia (SNe-Ia)support a constant dark energy equation of state with w ∼ −
1. Measuring the evolution of w along the redshift is one of the mostdemanding challenges for observational cosmology. Aims.
We discuss the existence of a close relation for GRBs, named Combo-relation, based on characteristic parameters of GRBphenomenology such as the prompt intrinsic peak energy E p , i , the X-ray afterglow, the initial luminosity of the shallow phase L , therest-frame duration τ of the shallow phase, and the index of the late power-law decay α X . We use it to measure Ω m and the evolutionof the dark energy equation of state. We also propose a new calibration method for the same relation, which reduces the dependenceon 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) agood coverage of the observed (0.310) keV afterglow light curves. The fitting technique of the rest-frame (0.310) keV luminosity lightcurves represents the core of the Combo-relation. We separate the early steep decay, considered a part of the prompt emission, fromthe X-ray afterglow additional component. Data with the largest positive residual, identified as flares, are automatically eliminateduntil the p-value of the fit becomes greater than 0.3. Results.
We strongly minimize the dependency of the Combo-GRB calibration on SNe Ia. We also measure a small extra-Poissonianscatter of the Combo-relation, which allows us to infer from GRBs alone Ω M = . + . − . (1 σ ) for the Λ CDM cosmological model,and Ω M = . + . − . , w = − . + . − . for the flat-Universe variable equation of state case. Conclusions.
In view of the increasing size of the GRB database, thanks to future missions, the Combo-relation is a promising tool formeasuring Ω m with an accuracy comparable to that exhibited by SNe Ia, and to investigate the dark energy contribution and evolutionup to z ∼ Key words.
Cosmology: observations, Gamma-ray burst: general, Cosmology: dark energy
1. Introduction
Gamma-ray bursts (GRBs) are observed in a wide range of spec-troscopic and photometric redshifts, up to z ∼ z universe, in termsof investigating the re–ionization era, population III stars, themetallicity of the circumburst medium, the faint–end of galax-ies luminosity evolution (D’Elia et al. 2007; Robertson & Ellis2012; Macpherson et al. 2013; Trenti et al. 2013, 2015), and us-ing GRBs as cosmological rulers (e.g. Ghirlanda et al. 2004; Daiet al. 2004; Amati & Della Valle 2013). The latter works showthat GRBs, through the correlation between radiated energy orluminosity and the photon energy at which their ν F ( ν ) spec-trum peaks E p , i , admittedly with a lower level of accuracy, pro-vide results in agreement with supernovae Ia (SN Ia, Perlmutteret al. 1998, 1999; Schmidt et al. 1998; Riess et al. 1998), bary-onic acoustic oscillations (BAO, Blake et al. 2011), and cosmicmicrowave background (CMB) radiation (Planck Collaborationet al. 2013). 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 Send o ff print requests to : e-mail: [email protected] e-mail: [email protected],[email protected] acceleration. Measuring the equation of state (EOS, ω = p /ρ ,with p the pressure and ρ the density of the dark energy) isone of the most di ffi cult tasks in observational cosmology to-day. Current data (Suzuki et al. 2012; Planck Collaboration et al.2015) suggest that w ∼ − w a ∼
0, the expected valuesfor the cosmological constant. Although these results are prob-ably su ffi cient to exclude a very rapid evolution of dark energywith z , we cannot yet exclude that it may evolve with time, asoriginally proposed by Bronstein (1933). In principle, with SNeIa we can push our investigation up to z ≈ . z ∼ − . z = . z and their very high luminosities, GRBs are a class of ob-jects suitable to exploring the trend of dark energy density withtime (Lloyd & Petrosian 1999; Ramirez-Ruiz & Fenimore 2000;Reichart et al. 2001; Norris et al. 2000; Amati et al. 2002;Ghirlanda et al. 2004; Dai et al. 2004; Yonetoku et al. 2004;Firmani et al. 2006; Liang & Zhang 2006; Schaefer 2007; Amati a r X i v : . [ a s t r o - ph . C O ] A ug . Izzo, M. Muccino, E. Zaninoni, et al.: Cosmological parameters and GRBs et al. 2008; Capozziello & Izzo 2008; Dainotti et al. 2008;Tsutsui et al. 2009; Wei et al. 2014). There are two complicationsconnected with these approaches. First, the correlations are al-ways calibrated by using the entire range of SNe Ia up to z = . ff erent extra-scatters published inMargutti et al. 2013), even when a large GRB dataset is used. Inthis work we present a method that can override the latter andminimize the former.Very recently Bernardini et al. (2012) and Margutti et al.(2013) (hereafter B12 and M13, respectively) have published aninteresting correlation that connects the prompt and the after-glow emission of GRBs (see Fig. 1). This relation strongly linksthe X-ray and γ -ray isotropic energy with the intrinsic peak en-ergy, E X , iso ∝ E . ± . γ, iso / E . ± . p , i , unveiling an interesting con-nection between the early, more energetic component of GRBs( E γ, iso and E p , i ) and their late emission ( E X , iso ), opportunely fil-tered for flaring activity. In addition, the intrinsic extra scatteris very small σ E X , iso = . ± .
06 (1 σ ). Starting from the re-sults obtained by B12 and M13, and combining them with thewell-known E p , i − E iso relation (Amati et al. 2002), we presenthere a new correlation involving four GRB parameters, whichwe named the Combo-relation. This new relation is then used tomeasure Ω m and to explore the possible evolution of w of EOSwith the redshift by using as a “candle” the initial luminosity L of the shallow phase of the afterglow, which we find to be strictlycorrelated with quantities directly inferred from observations.The paper is organized as follows. In Sect. 2 we describein detail how we obtain the formulation of the newly proposedGRB correlation, and present the sample of GRBs used to testour results, the procedure for the fitting of the X-ray afterglowlight curves, and finally, the existence of the correlation assum-ing a standard cosmological scenario. Then, in Sec. 3, we dis-cuss the use of the relation as cosmological parameter estimator,which involves a calibration technique that does not require theuse of the entire sample of SNe Ia, as was often done in previoussimilar works. In Sec. 4, we test the use of GRBs estimating themain cosmological parameters, as well as the evolution of thedark energy EOS. Finally, we discuss the final results in the lastsection.
2. The Combo-relation
We present here a new GRB correlation, the Combo-relation,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 analyticalformulation of the X-ray afterglow component given in Ru ffi niet al. (2014) (hereafter R14).The three-parameter scaling law reported in B12 and M13can be generally written as E X , iso ∝ E βγ, iso E δ p , i , (1)where E X , iso is the isotropic energy of a GRB afterglow in therest-frame (0 . E γ, iso is the isotropic energy of a GRB prompt emission, and E p , i is theintrinsic spectral peak energy of a GRB. Since E X , iso and E γ, iso are both cosmological-dependent quantities, we reformulate this Fig. 1.
The E X , iso – E γ, iso – E p , i correlation proposed by Bernardini et al.(2012) and Margutti et al. (2013) (courtesy R. Margutti) . correlation in order to involve only one cosmological-dependentquantity.The right term of Eq. 1 can be rewritten using the well-known formulation of the Amati relation, E γ, iso ∝ E η p , i , whichprovides E X , iso = A E γ p , i , (2)where γ = η × β − δ and A is the normalization constant. Sincethe Amati relation is valid only for long bursts, in the follow-ing discussion we will exclude short GRBs with a rest-frame T duration smaller than 2 s, and short bursts with “extendedemission” (Norris & Bonnell 2006), or “disguised short” GRBs(Bernardini et al. 2007; Caito et al. 2010), which have hybridcharacteristic between short and long bursts.To rewrite Eq. 2 for cosmological purposes, we have calcu-lated the total isotropic X-ray energy in the rest-frame (0 . L ( t (cid:48) a ), ex-pressed as a function of the cosmological rest-frame arrival time t (cid:48) a , over the time interval ( t (cid:48) a , , t (cid:48) a , ). This luminosity L ( t (cid:48) a ) is ob-tained by considering four steps (see e.g. Appendix A and Pisaniet al. 2013).It is well known that the X-ray afterglow phenomenologycan be described by the presence of an additional componentemerging from the soft X-ray steep decay of the GRB promptemission, and characterized by a first shallow emission, usu-ally named the plateau, and a late power-law decay behaviour(Nousek et al. 2006; Zhang et al. 2006; Willingale et al. 2007). Inaddition, many GRB X-ray light curves are characterized by thepresence of large, late–time flares, whose origin is very likely as-sociated with late–time activity of the internal engine (Marguttiet al. 2010). Since their luminosities are much lower than theprompt one, we exclude X–ray flares from our analysis via alight–curve fitting algorithm, which will be explained later in thetext. We then make the assumption that the E X , iso quantity refersonly to the component whose X-ray luminosity L ( t (cid:48) a ) is given bythe phenomenological function defined in R14 L ( t (cid:48) a ) = L (cid:32) + t (cid:48) a τ (cid:33) α X , (3)where L , τ , and α X are, respectively, the luminosity at t (cid:48) a =
2. Izzo, M. Muccino, E. Zaninoni, et al.: Cosmological parameters and GRBs
Therefore, if we extend the integration time interval to t (cid:48) a , → t (cid:48) a , → + ∞ , the integral of the function L ( t (cid:48) a ) in Eq. 3 gives E X , iso = (cid:90) + ∞ L ( t (cid:48) a ) dt (cid:48) a = L τ | + α X | , (4)with the requirement that α X < −
1. This condition is necessaryto exclude divergent values of E X , iso computed from Eq. (4), for t (cid:48) a , → + ∞ . It is worth noting that light curves providing values α X > − / or, in principle, could be pollutedby a late flaring activity, resulting in a less steep late decay.Considering Eqs. 2 and 4, we can finally formulate the fol-lowing relation between GRB observables L = AE γ p , i (cid:32) τ | + α X | (cid:33) − , (5)which we name the Combo-relation. At first sight, Eq. 5 suggeststhe existence of a physical connection between specific physicalproperties of the afterglow and prompt emission in long GRBs.A two-dimensional fashion of the correlation in Eq. 5 in loga-rithmic units can be written aslog (cid:32) L erg / s (cid:33) = log (cid:32) A erg / s (cid:33) + γ (cid:34) log (cid:32) E p , i keV (cid:33) − γ log (cid:32) τ/ s | + α X | (cid:33)(cid:35) , (6)where the set of parameters in the square brackets has the mean-ing of a logarithmic independent coordinate, and in the followingwill be expressed as log (cid:104) X ( γ, E p , i , τ, α X ) (cid:105) .We have tested the reliability of Eqs. 5 and 6 by building asample 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 X-ray afterglow lightcurve from the early decay ( t (cid:48) a ∼
100 s, when present) un-til late emission ( t (cid:48) a ∼ –10 s).We start the analysis by computing the rest-frame (0 . χ statis-tic, which eliminates the flaring part (see e.g. Appendix B andMargutti et al. 2011; Zaninoni 2013, and M13, for details). Atotal of 60 GRBs are found, whose distribution in the Combo-relation plane is shown in Fig. 2, where the value of the luminos-ity L for each GRB is calculated from the flat Λ CDM scenario.The corresponding best-fit parameters, as well as the extra-scatter term σ ext , have been derived by following the procedureby D’Agostini (2005), and are respectively log (cid:2) A / (erg / s) (cid:3) = . ± . γ = . ± .
10, and σ ext = ± ffi cient is ρ S = .
92, while thep-value computed from the two-sided Student’s t -distribution, is p val = . × − .
3. Calibration of the Combo-relation
The lack of very nearby ( z ∼ .
01) GRBs prevents the possibilityof calibrating GRB correlations, as is usually done with SNe Ia.In recent years di ff erent methods have been proposed to avoidthis “circularity problem” in calibrating GRB relations (Kodamaet al. 2008; Demianski et al. 2012, and references therein). The (cid:225) (cid:225)(cid:225)(cid:225) (cid:225)(cid:225)(cid:225)(cid:225) (cid:225) (cid:225)(cid:225)(cid:225) (cid:225) (cid:225)(cid:225) (cid:225)(cid:225)(cid:225)(cid:225)(cid:225) (cid:225)(cid:225) (cid:225) (cid:225) (cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225)(cid:225) (cid:225)(cid:225)(cid:225)(cid:225) (cid:225) (cid:225)(cid:225) (cid:225)(cid:225) (cid:225)(cid:225)(cid:225) (cid:225)(cid:225)(cid:225) (cid:225)(cid:225) (cid:225)(cid:225) (cid:225)(cid:225) (cid:225)(cid:225) (cid:225) (cid:225)(cid:225) (cid:225)(cid:225)(cid:225) (cid:45) (cid:45) (cid:45) (cid:64) E p , i (cid:144) keV (cid:68) (cid:45) (cid:180) Log (cid:64)(cid:72) Τ (cid:144) s (cid:76)(cid:144)(cid:200) (cid:43)Α (cid:200)(cid:68) L og (cid:64) L (cid:144) (cid:72) e r g (cid:144) s (cid:76) (cid:68) Fig. 2.
Plot of the correlation considering the entire sample of 60 GRBs.The green empty boxes are the data of each of the sources, derivedas described in Appendix B, the solid black line is the best fit of thedata, while the dotted grey lines and the dashed grey lines correspond,respectively, to the dispersion on the correlation at 1 σ ex and 3 σ ex . (cid:87) m (cid:61) (cid:87) (cid:76) (cid:61) (cid:87) m (cid:61) . , (cid:87) (cid:76) (cid:61) (cid:87) m (cid:61) , (cid:87) (cid:76) (cid:61) (cid:87) m (cid:61) , (cid:87) (cid:76) (cid:61) (cid:87) m (cid:61) , (cid:87) (cid:76) (cid:61) (cid:87) m (cid:61) , (cid:87) (cid:76) (cid:61) (cid:45) (cid:45) Μ ob s (cid:45) Μ t h Fig. 3.
The residual distance modulus µ obs − µ th for di ff erent values ofthe density cosmological parameters Ω m and Ω Λ up to z = .
0. Weconsider the best fit to be the standard Λ CDM model, where Ω m = . Ω Λ = . , and H =
71 km / s / Mpc (black line). Union2 SNe Ia dataresiduals are shown in grey. The large spread (more than 1 magnitude)shown by µ at z = . z = .
145 (the two vertical dashed lines)where the scatter is almost 0.2 magnitudes is clearly evident. common approach uses an interpolating function for the distri-bution of luminosity distances (distance moduli) of SNe Ia withredshift, and then extending it to GRBs at higher redshifts. In thefollowing we introduce and describe an alternative method forcalibrating the Combo-relation (see also Ghirlanda et al. 2006)which consists of two steps: 1) we identify a small but su ffi cientsubsample of GRBs that lie at the same redshift, and then inferthe slope parameter γ from a best-fit procedure; 2) once we de-termine γ , the luminosity parameter A can be obtained from adirect comparison between the nearest, z = . ff erent from previous ones in that we do notuse the whole redshift range covered by SNe Ia, but we limit ourcalibration analysis to z = ff ect of the cosmol-ogy on the distance modulus of the calibrating SNe Ia is small(see Fig. 3).
3. Izzo, M. Muccino, E. Zaninoni, et al.: Cosmological parameters and GRBs
Table 1.
List of the five GRBs used for the determination of the slope parameter γ of the correlation. In the Col. 1 is shown the GRB name, in Col.2 the redshift z , in Col. 3 the intrinsic peak energy of the burst, in Col. 4 the initial flux of the afterglow additional component F in the commonrest-frame energy range (0.3 – 10) keV, in Col. 5 the time parameter ( τ ), in Col. 6 the late power-law decay index ( α X ).GRB z E p , i (keV) F (erg cm − s − ) τ (s) α X .
54 104 ±
24 (2 . ± . × − ± − . ± . . ±
15 (4 . ± . × − ± − . ± . .
544 273 ± . ± . × − ± − . ± . .
54 257 ±
41 (1 . ± . × − ± − . ± . .
542 146 ±
23 (1 . ± . × − ± − . ± . (cid:243) (cid:243) (cid:243) (cid:243)(cid:243) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:64) E p,i (cid:144) keV (cid:68) (cid:45) (cid:64)(cid:72) Τ (cid:144) s (cid:76)(cid:144)(cid:200) (cid:43)Α X (cid:200)(cid:68) L og (cid:64) F (cid:144) (cid:72) e r g (cid:144) c m (cid:144) s (cid:76) (cid:68) Fig. 4.
Plot of the correlation found for the sample of five GRBs lo-cated at the same redshift. The blue triangles are the data of each ofthe five sources, derived from the procedure in Appendix B. The solidline represents the best fit while the dashed line is the dispersion on thecorrelation at 1 σ ex . γ The existence of a subsample with a su ffi cient number of GRBslying at almost the same cosmological distance would, in prin-ciple, allow us to infer γ , overriding any possible cosmologicaldependence (assuming a homogeneous and isotropic universe).In our sample of 60 GRBs there is a small subsample of 5 GRBslocated at a very similar redshift, see Table 1. The di ff erence be-tween the maximum redshift of this 5–GRB sample ( z = . z = . .
015 in redshift and of 0 .
07 in distance modulus µ in thecase of the standard Λ CDM model. This very small di ff erence issu ffi cient for our purposes.To avoid any possible cosmological contamination, we donot consider the luminosity L as the dependent variable, but weconsider instead the energy flux F , which is related to the lu-minosity through the expression L = π d l F . This assumptiondoes not influence the final result, since d l is almost the same forthe 5–GRB sample and, therefore, the 4 π d l term can be absorbedin the normalization constant. Consequently, we build the energyflux light curve for each GRB, and then, following the same pro-cedure described in Sec. 2, we perform a best-fit analysis of thissubsample of five GRBs using the maximum likelihood tech-nique. From the best fit we find a value of γ = . ± .
15, withan extra scatter of σ ext = . ± .
04, see Fig. 4. A with SNe Ia Among the considered 60 GRBs, the nearest one is GRB130702A at z = . Table 2.
List of the 5 SNe Ia selected from the Union2 sample(Amanullah et al. 2010; Suzuki et al. 2012) and used for the calibrationof the GRB correlation. In the first column it is shown the name of theSN, in the second column the redshift and in the third the distance mod-ulus obtained from the light curve fitting with the SALT2 SED method(Guy et al. 2007).SN z µ . ± . . ± . . ± . . ± . . ± . is clear that at these redshifts ( z = . ff set provided bySNe Ia is much smaller than the value inferred from SNe Ia atlarger redshifts, see Fig. 3. In this light we have selected Union2SNe Ia (Amanullah et al. 2010; Suzuki et al. 2012) with red-shift between z = .
143 and z = . (cid:104) µ (cid:105) = . ± .
27, which will be used for thefinal calibration. The parameter A can be obtained inverting Eq.6, and considering L = π dl F :log (cid:32) A erg / s (cid:33) = (cid:32) d l cm (cid:33) + log 4 π + log (cid:32) F erg / cm / s (cid:33) + − γ log X ( γ, E p . i , τ, α X ) . (7)The generic SN distance modulus µ can be directly related to theluminosity distance d l by µ = + (cid:32) d l Mpc (cid:33) = − . + (cid:32) d l cm (cid:33) , (8)where the last equality takes into account the fact that d l is ex-pressed in cm. Substituting this last expression for log ( d l ) in Eq.7 we obtainlog (cid:32) A erg / s (cid:33) =
25 ( (cid:104) µ (cid:105) + . + ψ ( γ, E p . i , τ, α X , F ) , (9)where the term ψ ( γ, E p . i , τ, α X , F ) comprises the last three termson the right hand side of Eq. 7. Substituting the quantities forGRB 130702A in Eq. 9, and the value of (cid:104) µ (cid:105) previously obtained,we infer a value for the parameter log[ A / (erg / s)] = . ± . σ ext value found above intoaccount.
4. Cosmology with the Combo-relation
We now discuss the possible use of the proposed GRB Combo-relation to measure the cosmological constant and the mass den-sity, as well as their evolution with redshift z . The possibility of
4. Izzo, M. Muccino, E. Zaninoni, et al.: Cosmological parameters and GRBs estimating the luminosity distance d l from the GRB observablequantities allows to us define a distance modulus for GRBs, andthen its uncertainty, as µ GRB = − . + (cid:104) A − ψ ( γ, E p , i , τ, α X , F ) (cid:105) , (10)where A = log[ A / (erg / s)]. The quantity µ GRB can be directlycompared with the theoretical cosmological expected value µ th ,which depends on the density parameters Ω m and Ω Λ , the curva-ture term Ω k = − Ω m − Ω Λ , and the Hubble constant H µ th = + d l ( z , Ω m , Ω Λ , H ) . (11)The luminosity distance d l is defined as d l = cH (1 + z ) √| Ω k | sinh (cid:20)(cid:82) z √| Ω k | dz (cid:48) √ E ( z (cid:48) , Ω m , Ω Λ , w ( z )) (cid:21) , Ω k > (cid:82) z dz (cid:48) √ E ( z (cid:48) , Ω m , Ω Λ , w ( z )) , Ω k = √| Ω k | sin (cid:20)(cid:82) z √| Ω k | dz (cid:48) √ E ( z (cid:48) , Ω m , Ω Λ , w ( z )) (cid:21) , Ω k < E ( z , Ω m , Ω Λ , w ( z )) = Ω m (1 + z ) + Ω Λ (1 + z ) + w ( z )) + Ω k (1 + z ) (see e.g. Goobar & Perlmutter 1995). In the following wefix the Hubble constant at the recent value inferred from low-redshift SNe Ia, corrected for star formation bias, and calibratedwith the LMC distance (Rigault et al. 2014): H = . ± . µ GRB were computedconsidering an observed term, σµ obs , which takes into accounteach uncertainty on the observed quantities of the Combo-relation, e.g. F , τ, α , and E p , i , and a “statistical” term, σµ rel ,which takes into account the uncertainties on the parameters ofthe Combo-relation, A and γ , and the weight of the extra scattervalue σ ext . The final uncertainty on each single GRB distancemodulus σµ GRB = (cid:113) ( σµ obs ) + ( σµ rel ) (13)allows us to build the Combo-GRB Hubble diagram (see alsoIzzo et al. 2009), which is shown in Fig. 5. It is possible to quan-tify the reliability of any cosmological model with our sampleof 60 GRBs, which represents a unique dataset from low red-shift ( z = . z = . χ = (cid:88) i = (cid:2) µ GRB , i ( z , A , ψ ) − µ th ( z , Ω m , Ω Λ , w ( z )) (cid:3) σµ GRB , i , (14)where µ GRB ( z , A , ψ ), µ th ( z , Ω m , Ω Λ , w ( z )) , and σµ GRB are respec-tively defined in Eqs. 10, 11, and 13. The cosmology is includedin the quantity µ th ( z , Ω m , Ω Λ , H ), which we allow to vary. Todetermine the best configuration of parameters that most closelyfits the distribution of GRBs in the Hubble diagram we maxi-mize the log-likelihood function, − e χ ), which is equivalentto the minimization of the function defined in Eq. 14. Λ CDM case
In the Λ CDMmodel, the energy function E ( z , Ω m , Ω Λ , w ( z )) ischaracterized 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 matterand cosmological constant density parameters, also obtaining inthis way an estimate of the curvature term. We obtain that GRBsalone provide Ω m = . + . − . , see also Fig. 6. D i s t a n ce m odu l u s Μ Fig. 5.
The Combo-GRB Hubble diagram. The black line represents thebest fit for the function µ ( z ) as obtained by using only GRBs and for thecase of the Λ CDM scenario. (cid:87) m (cid:87) (cid:76) Fig. 6. σ ( ∆ χ = .
3) confidence region in the ( Ω m , Ω Λ ) plane (leftpanel) and in the ( Ω m , w ) plane (right panel) for the Combo-GRB sam-ple (dark blue) , and for the total (60 observed +
300 MC simulatedGRBs, (light blue) . w case In a flat Universe (
Ω = Ω m + Ω Λ =
1) with a constant valueof w di ff erent from the standard value w = −
1, we can pro-vide useful constraints for alternative dark energy theories. Inthis case, we only vary the matter density and the dark energyequations of state, obtaining an estimate of the density mat-ter of Ω m = . + . − . and of a dark energy EOS parameter w = − . + . − . . w ( z )An interesting case–study consists of a time-evolving dark en-ergy EOS in a flat cosmology, since the evolution of w ( z ) canbe directly studied with GRBs at larger redshifts. We consideran analytical formulation for the evolution of w with the red-
5. Izzo, M. Muccino, E. Zaninoni, et al.: Cosmological parameters and GRBs (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:87) m w Fig. 7. σ ( ∆ χ = .
3) confidence region in the ( Ω m , w ) plane (rightpanel) for the Combo-GRB sample (dark blue) , and for the total of 60observed +
300 MC simulated GRBs (light blue) . The black dashed linerepresents the 1- σ confidence region obtained using the recent Union2.1 SNe Ia sample (Suzuki et al. 2012). shift, which was proposed by Chevallier & Polarski (2001) andLinder (2003) (CPL), and where the w ( z ) can be parameterizedby w ( z ) = w + w a z + z . (15)The CPL parameterization implies that for large z the w ( z )term tends to the asymptotic value w + w a . Using a sample ex-tending at large redshifts, e.g. GRBs, will allow a better estimateof these parameters since the e ff ects of a varying w ( z ) on the dis-tance modulus are more evident at redshift z ≥
1. The GRB dataprovide a best-fit result ( w = − . + . − . , w a = . + . − . ). The current sample of GRBs that satisfies the Combo-relation isquite limited (60 bursts), when compared to the sample in Amati& Della Valle (2013) ( ∼
200 bursts) or the SNe sample in theUnion 2.1 release (Suzuki et al. 2012). A more numerous samplecan help to understand whether the Combo-relation can providebetter 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 GRBssatisfying the Combo-relation. This value comes from the ex-pected number of GRBs detected in five years of operations ofcurrent (Swift) and future (SVOM (Gotz et al. 2009), and LOFT(Feroci et al. 2012)) missions dedicated to observing GRBs.First, we fitted the log-normal distributions of the 60 ob-served z , E p , i , τ , and | α + | , f (log ξ ) = √ πσ ξ exp − (cid:16) log ξ − µ ξ (cid:17) σ ξ , (16) where ξ = z , E p , i , τ , and | α + | , and we found the following meanvalues and dispersions: µ z ± σ z = . ± . µ E p , i ± σ E p , i = . ± . µ τ ± σ τ = . ± .
65, and µ | α + | ± σ | α + | = − . ± . X ( γ, E p . i , τ, α )and we generated the initial luminosity log L = γ log X + A from the frequency distribution f (log L ) = √ πσ ext exp − (cid:0) log L − γ log X − A (cid:1) σ ext , (17)assuming that the Combo-relation is independent of the redshiftand considering its extra-scatter σ ext . The values of γ , A , and σ ext are reported in Section 3. Finally, to complete the set of pa-rameters necessary to compute the distance modulus of the sim-ulated sample of GRBs from each pair (log L , z ), we generatedthe corresponding log F ( µ F ± σ F = − . ± . Ω m , Ω Λ ) willimprove by using a larger sample of 360 GRBs (the real sam-ple of 60 GRBs observed and a MC-simulated sample of 300GRBs, described above). The improvement on the constraintson Ω m and Ω Λ is clear: the uncertainties on the density parame-ters improve considerably ( Ω m = . + . − . for the Λ CDM case, w = − . + . − . for the variable w case), as is also clear in thecontour plots shown in Figs. 6 and 7. In order to compare the Combo-GRB sample results, we alsoconsider the following datasets:- the measurements of the baryon acoustic peaks A obs = (0 . ± . , . ± . , . ± . z BAO = (0 . , . , .
73) in the galaxy cor-relation function as obtained by the WiggleZ dark energySurvey (Blake et al. 2011). The BAO peak is defined as(Eisenstein et al. 2005) A BAO = (cid:34) cz BAO H r ( z BAO , Ω m , Ω Λ , w ( z )) E ( z , Ω m , Ω Λ , w ( z )) (cid:35) / (cid:113) Ω m H cz BAO , (18)where r ( z , Ω m , Ω Λ , w ( z )) is the comoving distance. The best-fit cosmological model is determined by the minimization ofthe corresponding chi-squared quantity χ BAO = (cid:88) i = ( A BAO − A ( obs , i ) ) T C − ( A BAO − A ( obs , i ) ) , (19)where C − is the inverse covariance matrix of the measure-ments of the WiggleZ survey (Blake et al. 2011).- the measurement of the shift parameter R obs = . ± . R quantity is the least cosmo-logical model-dependent parameter (particularly from H )that can be extracted from the analysis of the CMB (Wang& Mukherjee 2006) and is defined as R = (cid:112) Ω m (cid:90) z rec dz (cid:48) E ( z (cid:48) , Ω m , Ω Λ , w ( z )) , (20)
6. Izzo, M. Muccino, E. Zaninoni, et al.: Cosmological parameters and GRBs
GRBGRB sim300SNeBAOCMB (cid:87) m (cid:87) (cid:76) Fig. 8. σ ( ∆ χ = .
3) confidence region in the ( Ω m , Ω Λ ) plane for theobserved GRB sample ( blue ), with the inclusion of the MC–simulated300 GRBs sample ( cyan ), and with the samples of SNe ( grey ), BAOs( red ), and CMBs ( green ). The dashed line represents the condition ofthe Flat Universe Ω m + Ω Λ = where z rec is the redshift of the recombination. The best-fitcosmological model is determined by the minimization ofthe corresponding chi-squared term χ CMB = ( R obs − R ) σ R obs . (21)We use a grid-search technique to vary the values of the cos-mological which are used to solve numerically the chi-squaredequations defined above. Every dataset provides the respectivedistribution of the cosmological parameters, so when combiningdatasets we simply determine the best-fit model as the sum of thesingle χ i values for any pair of parameters and for the combinedvalues, χ tot = χ GRB + χ S Ne + χ BAO + χ CMB , (22)and make the final plots using the Mathematica software suite.Cosmological parameter uncertainties were estimated from sin-gle and combined χ statistics for each dataset. In Fig. 8 we showthe parameter spaces with the contour at the 1- σ ( ∆ χ = . Λ CDMcase. We note that in all the datasets and fits, the Hubble constantis fixed to the value of H = . ± . Λ CDMcase, we obtain a matter density value of Ω m = . + . − . , whichshows no strain with the Combo-GRB results and the simulatedone, see Fig. 8.
5. Discussions and conclusions
In this paper we have presented the “Combo” relation, a newtool for GRB cosmology. This relationship provides a very close http: // / mathematica / link between prompt and afterglow parameters and it is charac-terized by a small intrinsic scatter, which makes this correlationvery suitable for cosmological purposes. We recognize the fun-damental role of the Swift -XRT (Gehrels et al. 2004; Burrowset al. 2005) which, thanks to its ability to slew very rapidly to-ward the location of a GRB event, provides real-time and de-tailed data of GRB afterglow light curves, whose evolution is atthe base of the proposed Combo-relation. From our analysis thefollowing results emerge: – the proposed two–step calibration of the Combo–GRB rela-tion greatly minimizes the dependence on SNe Ia; – GRBs data alone provide for the Λ CDM case Ω m = . + . − . , see Fig. 8; – a recent paper (Milne et al. 2015) highlights the existenceof an observational bias (a systematic di ff erence in the ve-locity of SNe Ia ejecta, which is reflected in their curves),potentially a ff ecting the measurements of cosmological pa-rameters obtained with SNe Ia. On the basis of our resultswe conclude that given the current accuracy of GRB mea-surements we cannot exclude, within the errors, that an e ff ectlike this is at play; however, this e ff ect should not change theconclusions derived from SNe-Ia observations; – the launch of advanced and more sensitive detectors, suchas the incoming SVOM (Gotz et al. 2009) and the proposedLOFT (Feroci et al. 2012) missions (and the expected Swiftoperations in the near future), will dramatically increase thenumber of GRBs in the dataset. In five years of operationof the SVOM mission alone, we expect to reach a “goodenough” sample of 300 GRBs. With a Monte Carlo simu-lated sample of 300 GRBs, we will significantly improve theaccuracy of Ω m measurement, up to Ω m = . + . − . , whichis comparable with type Ia SNe (Suzuki et al. 2012); – by using the CPL analytical parameterization, adopted tostudy the evolution of the dark energy EOS (see Eq. 15), wefind Ω m = . + . − . , w = − . + . − . , and w a = . + . − . ; – the analysis of a combined (SNe + BAO + CMB) dataset con-firms that the increasing size of the GRB sample will im-prove the accuracy of the measurement of the Ω m parameterand in particular of the evolution of w up to z ∼ – the analytical expression of the Combo-relation provides anexplicit close link, characterized by a small intrinsic scatter,between the prompt and the afterglow GRB emissions. Thispoints out the existence of a physical connection between theprompt and the afterglow emissions, which represents a newchallenge for GRB models. Acknowledgements.
We thank the referee for her / his constructive commentswhich have improved the paper. EZ acknowledges the support by theInternational Cooperation Program CAPES-ICRANet financed by CAPES -Brazilian Federal Agency for Support and Evaluation of Graduate Educationwithin the Ministry of Education of Brazil. This work made use of data suppliedby the UK Swift
Science Data Centre at the University of Leicester.
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Appendix A: Computation of the rest-frame . – keV luminosity L ( t (cid:48) a ) The rest-frame 0 . L ( t (cid:48) a ) was obtained byconsidering four steps.(1) We obtained the Swift -XRT flux light curves in the observed0 . .(2) We transformed the observed flux f obs from the observed en-ergy band 0 . . N ( E ) ∼ E − γ , with Galactic and intrinsic column densitiesobtained from the H I radio map (Kalberla et al. 2005) andfrom the best fit of the total afterglow spectrum, respectively.By using the photon indexes inferred for each time interval,the rest-frame flux light curve f r f is given by f r f = f obs (cid:82) / (1 + z )0 . / (1 + z ) N ( E ) EdE (cid:82) . N ( E ) EdE = f obs (1 + z ) γ − . (A.1)(3) We transformed the observed time t a into the rest-frame timeby correcting for z t (cid:48) a = t a / (1 + z ) . (A.2)(4) We defined the isotropic luminosity as L = π d l f r f . (A.3) Appendix B: Determination of the sample andverification of the correlation
To obtain the parameters involved in the Eq. 5, we needed to se-lect an adequate sample of GRBs, to fit their X-ray light curves,and to collect or to calculate their E p , i values. The selection cri-teria have already been delineated in Sec. 2.The entire procedure works in the rest-frame 0 . language, andthe fitting routine used is MPFIT (Markwardt 2009), which isbased on the non-linear least squares fitting. First, the procedurefits the complete light curve, then it eliminates at every iterationthe data point with the largest positive residual, until it obtains afit with a p-value greater than 0 .
3. To fit the light curves consid-ered in luminosity units (erg / s), we use the composite function(R14):1. a power law for the early steep decay L ( t ) = L p (cid:32) t (cid:48) a (cid:33) − α p , (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. ?? for the case of GRB 060418. After the fitting procedure, weselect only the GRBs with α X < −
1, a condition necessary forthe convergence of the integral in Eq. 4. The final total sample,summarized in Table B.1, consists of 60 GRBs. http: // / burst analyser / Interactive Data Language, http: // / language / en-US / ProductsServices / IDL.aspx http: // purl.com / net / mpfit8. Izzo, M. Muccino, E. Zaninoni, et al.: Cosmological parameters and GRBs Table B.1.
Long bursts with α X < − z (second column), intrinsicpeak energy E p , i (third column), flux in the 0 . F (fourth column), the time parameter τ (fifth column), and the late power-lawdecay index α X (sixth column). All errors are at the 1- σ confidence level.GRB z E p , i (keV) F (erg cm − s − ) τ (s) α X . ±
110 (2 . ± . × − ± − . ± . .
198 415 ±
111 (3 . ± . × − ± − . ± . .
346 539 ±
200 (2 . ± . × − ± − . ± . .
53 285 ±
34 (5 . ± . × − ± − . ± . .
489 572 ±
143 (6 . ± . × − ± − . ± . .
21 105 ±
21 (5 . ± . × − ± − . ± . .
43 279 ±
28 (1 . ± . × − ± − . ± . .
54 104 ±
24 (2 . ± . × − ± − . ± . .
84 473 ±
155 (5 . ± . × − ± − . ± . . ±
102 (1 . ± . × − ± − . ± . .
467 475 ±
47 (1 . ± . × − ± − . ± . .
314 1289 ±
153 (3 . ± . × − ± − . ± . .
145 1013 ±
160 (3 . ± . × − . ± . − . ± . .
937 1261 ±
65 (6 . ± . × − . ± . − . ± . . ±
30 (4 . ± . × − ± − . ± . . ±
55 (3 . ± . × − . ± . − . ± . .
036 1691 ±
226 (2 . ± . × − ± − . ± . .
591 1741 ±
227 (2 . ± . × − ± − . ± . .
35 1470 ±
180 (1 . ± . × − ± − . ± . .
689 184 ±
18 (5 . ± . × − ± − . ± . .
692 95 ±
23 (3 . ± . × − ± − . ± . . ±
15 (4 . ± . × − ± − . ± . . ±
52 (6 . ± . × − ± − . ± . .
77 505 ±
34 (1 . ± . × − ± − . ± . .
547 1149 ±
166 (5 . ± . × − ± − . ± . .
608 1567 ±
384 (4 . ± . × − ± − . ± . . ±
55 (6 . ± . × − ± − . ± . .
544 273 ± . ± . × − ± − . ± . .
109 971 ±
390 (4 . ± . × − ± − . ± . .
54 257 ±
41 (1 . ± . × − ± − . ± . .
971 55 ±
20 (8 . ± . × − ± − . ± . .
71 280 ±
190 (3 . ± . × − ± − . ± . .
752 230 ±
66 (1 . ± . × − ± − . ± . .
49 53 . ± . . ± . × − ± − . ± . .
542 146 ±
23 (1 . ± . × − ± − . ± . .
44 259 ±
34 (1 . ± . × − ± − . ± . .
727 289 ±
46 (1 . ± . × − ± − . ± . .
46 231 ±
21 (4 . ± . × − ± − . ± . .
77 421 ±
14 (8 . ± . × − ± − . ± . .
714 58 . ± . . ± . × − ± − . ± . .
728 515 ±
22 (2 . ± . × − ± − . ± . .
671 198 ±
11 (7 . ± . × − ± − . ± . .
97 303 ±
65 (7 . ± . × − ± − . ± . . ±
14 (1 . ± . × − ± − . ± . . ±
15 (2 . ± . × − ± − . ± . .
023 194 ±
26 (2 . ± . × − ± − . ± . .
757 1294 ±
190 (1 . ± . × − ± − . ± . .
297 129 ± . ± . × − ± − . ± . . ± . ± . × − ± − . ± . .
27 2063 ±
101 (4 . ± . × − ± − . ± . .
092 912 ±
133 (2 . ± . × − ± − . ± . .
006 186 ±
31 (1 . ± . × − ± − . ± . .
155 192 ± . ± . × − ± − . ± . .
145 14 . ± . . ± . × − ± − . ± . . ± . ± . × − ± − . ± . .
295 406 ±
22 (8 . ± . × − ± − . ± . .
686 548 ±
83 (3 . ± . × − ± − . ± . .
042 222 ±
37 (2 . ± . × − ± − . ± . .
73 448 ±
22 (1 . ± . × − ± − . ± . . ± . ± . × − ± − . ± .04