A cosmological independent calibration of the Ep,i-Eiso correlation for Gamma Ray Bursts
aa r X i v : . [ a s t r o - ph . C O ] M a r Astronomy&Astrophysicsmanuscript no. GRB˙amati˙cosmo c (cid:13)
ESO 2018November 13, 2018
A cosmological independent calibration of the E p , i - E iso correlationfor Gamma Ray Bursts S. Capozziello and L. Izzo , , Dipartimento di Scienze Fisiche, Universit`a di Napoli ”Federico II” and INFN Sez. di Napoli, Compl. Univ. Monte S. Angelo, Ed.N, Via Cinthia, I-80126 Napoli, Italy ICRANet and ICRA, Piazzale della Repubblica 10, I-65122 Pescara, Italy. Dip. di Fisica, Universit`a di Roma ”La Sapienza”, Piazzale Aldo Moro 5, I-00185 Roma, Italy.Preprint online version: November 13, 2018
ABSTRACT
Aims.
The relation connecting the emitted isotropic energy and the rest-frame peak energy of the ν F ν spectra of Gamma-Ray Bursts(the Amati relation), strictly depends on the cosmological model, so we need a method to obtain an independent calibration of it. Methods.
Using the Union Supernovae Ia catalog, we obtain a cosmographic luminosity distance in the y -redshift and demonstrate thatthis parametrization approximates very well the fiducial standard comsomlogical model Λ CDM. Furthermore, by this cosmographicluminosity distance d l , it is possible to achieve the Amati relation independent on the cosmological model Results.
The cosmographic Amati relation that we obtain agrees, in the errors, with other cosmological-independent calibrationsproposed in the literature.
Conclusions.
This could be considered a good indication in view to obtain standard candles by Gamma-Ray Bursts
Key words.
Gamma rays : bursts - Cosmology : cosmological parameters - Cosmology : distance scale
1. Introduction
Supernovae Ia (SNeIa) are considered accurate and reliable stan-dard candles, (Phillips et al.). In recent years, their use as cos-mological distance indicators have led to the puzzling discov-ery that the Universe is in a phase of accelerated expansion,(Riess et al. 1998; Perlmutter et al. 1999). This feature has alsoled to the revision of the standard cosmological model, lead-ing to what is known today as the Λ CDM concordance model,see e.g. (Ostriker and Steinhardt 1995). However it is not pos-sible to observe these objects very far in the Universe. Themost distant Supernova Ia was observed at a redshift of z ∼ ∼ / or spectroscopicproperties of GRBs themselves. In the scientific literature thereare several of these relations, (Schaefer 2006). One of these isthe Amati relation, (Amati 2002), which relates the isotropic en-ergy emitted by a GRB with the peak energy in the rest-frame of the ν F( ν ) electromagnetic spectrum of a GRB. This relationhas already been widely used to constraining the cosmologicaldensity parameter (Amati et al. 2008), with quite remarkable re-sults. However, there is still not a physical link between this cor-relation and the mechanisms underlying the production and theemission of a GRB. The basic emission process of a GRB is verylikely not unique, so it is not easy to explain, from a physicalpoint of view, such a relation. Recently it has been suggested thatthe Amati relation could depend strongly on the satellite mea-surements used for detection and the observation of each GRB(Butler et al. 2007). However this hypothesis has been rejectedrecently, (Amati et al. 2008), since the relation seems to be ver-ified regardless of the satellite considered for the observationsand detection.Although not supported by self-consistent physical moti-vations, it is a phenomenological relation which could be ex-tremely useful for cosmological considerations. However, aproblem related with such a relation is that it must be calibratedindependently of the considered cosmological model. In order tocompute the energy emitted from an astrophysical object at a cer-tain redshift z , we need, as a matter of fact, a measurement of thebolometric flux and the distance of the same object. For the firstquantity, we follow the idea outlined by (Schaefer 2006) : onecan obtain a very precise measurement of the bolometric fluenceemitted by a GRB from the observed fluence, the integrated fluxin the observation time and the spectral model that best fits thespectral energy distribution of each GRB. However, the distancedepends on the considered cosmological model. People usuallyadopt the standard Λ CDM model, with fixed values of the den-sity parameter Ω i . This procedure leads to the so-called circu-larity problem when the Amati relation is used to standardizeGRBs. For this reason we need a cosmology-independent cali-bration of the relation. S. Capozziello and L. Izzo: A cosmological independent calibration of the E p , i - E iso correlation for Gamma Ray Bursts Recently, it was released a calibration with SNeIadata by using di ff erent numerical interpolation methods(Liang et al. 2008); the results seem very reliable to address cos-mological issues by GRBs. In this work we shall take into ac-count a similar analysis: by taking into account SNeIa data fromthe cosmographic point of view (for a detailed description seee.g. (Weinberg 1972; Visser 2004)), it could be possible to ob-tain a calibration of the Amati relation. We will use results ob-tained from a cosmographic fit of a sample of SNeIa extendedup to very high redshift with the GRBs. The use of the cosmog-raphy to deduce the cosmological parameters from SNeIa waswidely discussed in the literature, (Visser 2007a), and the resultsare very close to that attained by other and more accurate analy-sis. Recently applications of cosmographic methods have takeninto account galaxy clusters (Capozziello et al. 2004) and GRB,(Capozziello & Izzo 2008; Vitagliano et al. 2010) but their reli-ability drastically fails at high redshifts. Indeed, the estimatesof the deceleration parameter q and of the jerk parameter j are usually achieved only at very low redshift and then any ex-trapolation could led to shortcomings and misleading results assoon as they are extended. However by an appropriate parame-terization of the redshift parameter, one can circumvent the prob-lem introducing a new redshift variable ranging from 0 and 1(Visser 2004). Let us consider the following quantity as the newredshift variable: y = z + z , (1)we obtain that the range of variation is between 0 and 1. In thisway, we can derive a luminosity distance by which we can obtainthe Amati relation suitable for cosmography.The layout of the paper is the following: in Sect. 2 we tacklethe cosmographic analysis considering the SNeIa Union sample.Results will be used to derive the luminosity distance for eachGRB and then we will fit the cosmographic Amati relation. InSect. 3 a discussion on how to extend the same relation is re-ported. We add further 13 GRBs (as of December 2009), com-puting the bolometric fluence and the peak energy for each ofthem and after we calculate the cosmographic parameters usingthe new relation. Finally, we calculate the isotropic energy foreach GRB and then compute the best fit for the considered sam-ple of data (Sect. 4). Discussion and conclusions are reported inSec. 5.
2. Cosmographic analysis
The main purpose of this work consists in obtaining an Amatirelation independent of the adopted cosmological model. All weneed is a formulation of the luminosity distance d l as a func-tion of the redshift z . These two quantities are linked togethervia the scale factor a ( t ), which describes the expansion of theUniverse in a Friedmann-Lemaitre-Robertson-Walker cosmol-ogy. This means that we are assuming only homogeneity andisotropy but not the specific cosmological model, e.g. Λ CDMmodel. It is well known that we can obtain the function a ( t )from the Friedmann equations. These equations can be solvedonly if assumptions are made on dynamics and fluids filling theUniverse, that is choosing a cosmological model. We will re-lax this possibility assuming only cosmography in the sense de-scribed in (Weinberg 1972). Since the evolution of the luminos-ity distance is well known for small values of redshift, we canconsider the power series expansion of the scale factor. This nat-urally leads to an expression for the luminosity distance in powerseries terms too (Visser 2004; Capozziello et al. 2008): Table 1.
SNeIa cosmographic fit obtained by both the redshiftvariables z and y . Parameter value z -redshift Parameter value y -redshift a ± a ± b ± b ± c -0.8201 ± c -0.894 ± d ± d ± d l ( z ) = d H z ( + (cid:2) − q (cid:3) z − − q − q + j + k d H a z +
124 [2 − q − q − q + j (1 + q ) + s + k d H (1 + q ) a ] z + O ( z ) ) (2)where d H = c / H and H , q , j and s are known as theHubble constant, the deceleration, the jerk and the snap param-eters respectively. In order to obtain accurate measurements ofthe cosmographic parameters, we need to go up to large valuesof the redshift. This goal can be achieved by considering largedata sample as SNeIa (Visser 2004) and, eventually, GRBs.Here, we are interested in reconstructing the relation d l ( z ) bycosmographic methods in order to test correlations for GRBs . Inorder to achieve this goal, we will use the data sample of SNeIaUnion, (Kowalski et al. 2008) consisting of 307 supernovae upto redshift z ∼ .
7. By this data sample, it is possible to performa non-linear least-squares fit considering the empirical equationgiven by the distance modulus obtained from the expanded d l ( z ),that is: µ ( z ) = + ( d H [ z +
12 (1 − q ) z −
16 (1 + j + c ka H − q − q ) z +
124 (2 + j − q − q + c k (1 + q ) a H + j q − q + s ) z + O ( z )] ) . (3)In this work we are not interested in the estimate of the cos-mographic parameters but in using cosmography to constrain aGRB-energy relation. To this aim, we will use a custom equationfor the fit of the type µ ( z ) = + (5 / log 10) log( az + bz + cz + dz ) , (4)so we will compute only the parameters a , b , c , d . Once we havean estimate of these parameters, we could easily obtain the val-ues of the related cosmographic parameters. To obtain a betteranalysis, we can use a robust interpolation method of Levenberg-Marquardt type. The results of our data fitting are shown in Table1. The test of reliability of the fit has been done with a R -test, (Bevington et al. 2002), whose value is 0 . µ ( z ) shows aserious problem: for redshifts greater than ∼ . Capozziello and L. Izzo: A cosmological independent calibration of the E p , i - E iso correlation for Gamma Ray Bursts 3 rapidly, see Fig.1. This steep departure is due to the higher-order term, i.e. d , which has a decisive influence at high red-shift. This fact rules out a priori a possible supernova-calibrated µ ( z ) at high redshift. Such problems can be eliminated if weconsider a new variable for the redshift. It has been shown(Visser 2007a; Vitagliano et al. 2010) that the coordinates trans-formation y = z / (1 + z ) and, consequently, the power series ofthe luminosity distance provides a better extrapolation at highredshft, as well as better results for the parameters of the fit. Dueto this fact, we can perform a cosmographic analysis for the newdistance modulus µ ( y ), in analogy with what has been alreadydone for the µ ( z ). The new expression for the distance modu-lus, which takes into account the new redshift parameterization,becomes (Vitagliano et al. 2010): µ ( y ) = + ( log d H + log y −
12 ( q − y +
124 (21 − j + c ka H ) + q (9 q − y +
124 [15 + c ka H ( q − + j (8 q − − q + q − q + s ] y + O ( y )] ) (5)so we will consider a custom equation for the fit similar tothe previous one, used for the estimate of the µ ( z ) parameters.The results obtained with a non-linear fit are shown in Table 1,while in Figure 1 it is shown the trend of the distance modulusfor both the redshift variables considered.In the following we will consider the formulation for the dis-tance modulus in terms of the y -redshift in order to derive a cos-mographic Amati relation.
3. The data sample
As said in Introduction, in recent years the interest of astrophysi-cists and cosmologists has been attracted by the possibility ofusing GRBs as potential distance indicators. This interest is dueto the fact that most of the GRBs satisfy some correlations be-tween photometrical and spectroscopical observable quantities.Among the various existing correlations (for a review of thesesee e.g. (Schaefer 2006)), the Amati relation seems very attac-tive (Amati 2002). It relates the cosmological rest-frame ν F ( ν )spectrum peak energy E p , i with the equivalent isotropic radiatedenergy E iso . It was discovered based on BeppoSAX data and thenconfirmed also for the X-ray flashes (XRFs) (Lamb et al. 2004).It seems that it does not work for short GRBs. For this reasonthe relation could be used to discriminate among di ff erent GRBclasses.The possible origin of this correlation as due to detector se-lection e ff ects seems not consistent, nevertheless the large scat-ter in the normalization and the shift toward the Swift detectionthreshold (Butler et al. 2007). A recent study (Amati et al. 2009)has shown that the di ff erent E p , i - E iso correlations, obtained in-dependently from the detectors considered for the observations,are fully consistent each other, so the hypothesis of a strumental-dependent Amati relation seems to fail. Here we are going to expand the sample of GRBs reportedin (Amati et al. 2009) adding 13 GRBs and obtaining a sampleconsisting of 108 GRBs. Substantially we need to know the red-shift z , the observed peak energy E p , obs of the ν F ( ν ) spectrumand an estimate of the bolometric fluence S bolo for each GRB inthe sample. To derive the bolometric fluence S bol , we can use themethod outlined in (Schaefer 2006), where from the observedfluence and the spectral model, we can obtain an estimation of S bol via the following formula: S bol = S obs R / (1 + z )1 / (1 + z ) E φ dE R E max E min E φ dE (6)where φ is the spectral model considered for the spectral data fitand S obs is the fluence observed for each GRB in the respectivedetection band ( E min , E max ). In particular, for 6 of the 13 GRBsadded, we consider a cut-o ff power-law spectral model while forthe remaining 7 we use a band model (Band et al. 1993). In theTable 2, the spectral data for the 13 GRBs are shown. E p columnrefers to the measured peak energy. To obtain the peak energy inthe rest frame, we have to take into account the redshift of theGRB, then E p , i = E p (1 + z ). Once we have obtained the estimateof S bol for each GRB in the sample, the next step is to estimatethe isotropic energy from the well-known formula which relatesthe luminosity distance and the fluence, that is E iso = π d l S bol (1 + z ) − . (7)Note that the quantity (1 + z ) to obtain the value of an observablequantity in the rest-frame is equivalent, in the new redshift pa-rameterization, to use, instead, the correction 1 / (1 − y ). The valueof the luminosity distance which must enter in Eq.7 is what wegot previously from the cosmographic fit of the SNeIa. From thisfit, we obtained an estimate of the function µ ( y ); to go back tothe luminosity distance, we can use the following formula: d l ( y ) = µ ( y ) − (8)by which it is possible to compute the value of d l ( y ) for eachGRB in the sample.It is worth noticing that for values of y greater than ∼ µ ( y ) begins to increase slightly. This fact could leadto improper estimates of the isotropic energies emitted by GRBsat high redshift. If we consider an analogous curve referred toa fiducial standard Λ CDM cosmological model, we can quanti-tatively evaluate this deviation. In Figure 3, it is shown the de-viation of the curve µ ( y ), obtained by the cosmographic fit ofthe SNeIa and the one obtained by considering a Λ CDM modelwith values of the density parameters given by Ω ρ = .
27 and Ω Λ = .
73. The discrepancy from the fiducial Λ CDM modelseems quite small, but it has to be taken into account when wewill compute the cosmographic Amati relation.
4. The Cosmographic Amati relation
At this point we can calculate the parameters of the Amati rela-tion for the sample that we constructed previously. This relationis a correlation of type E iso = aE γ p , i ; however if we report it in alogarithmic basis, it reduces to the form:log E iso = A + γ log E p , i (9)so we can report our sample in a diagram log E iso - log E p , i and perform a linear fit of the data, with weights given by the S. Capozziello and L. Izzo: A cosmological independent calibration of the E p , i - E iso correlation for Gamma Ray Bursts z Μ y Μ Fig. 1.
Trends of the distance modulus for the z -redshift and for the y -redshift. Table 2.
Data for the 13 GRBs added to the old sample described in (Amati et al. 2009). Table shows: (1) the name of GRB, (2) thespectral model used for the fitting of the spectra, (3) the redshift, (4) the peak energy observed, (5) the softer spectral index, absentfor the cut-o ff power law spectral model, (6) the higher spectral index, (7) the observed fluence and (8) the detector band consideredfor the estimate of the fluence, (9) the GCN reference for the GRB, where we took the spectral data. GRB spec model z E p , o (keV) α β ( γ ) S obs (10 − ergs / cm ) band (keV) GCN (1) (2) (3) (4) (5) (6) (7) (8) (9)090516 CPL 4.109 190 ±
65 – -1.5 ± ± ±
56 – -1.1 ± ± ±
243 – -1.03 ± ± ± ± ± ±
16 – -1.53 ± ± ± ± ± ±
11 -1.26 ± ± ± ±
11 -0.696 ± ± ± ± ± ± ± ± ± ± ± ±
68 -0.93 ± ± ± ± ± ± ± ± ± ± ± data errors on both the quantities involved. An R -test providesan estimation of the reliability of the fit being R = σ , and thecorresponding covariance matrix are: A = . ± . γ = . ± .
117 (10) ( . − . − . . ) A comparison with the results obtained by di ff erent inter-polation methods (Liang et al. 2008) shows a slight discrepancybetween the parameters of the relation. This fact could be dueto the calibration in (Liang et al. 2008). It depends on the trendtraced by SNeIa, while the cosmographic analysis takes intoaccount the corrections due to physical parameters as q , j .Nevertheless the reason could be another: since the SNeIa sam-ple, used here to calibrate the Amati relation, is di ff erent fromthat in (Liang et al. 2008), where the authors adopted the cat-alog of 192 SNeIa discussed in (Wood-vasey et al. 2007). Thismeans that the slight di ff erence in the results could be due to thedi ff erent samples used for the calibration. In Fig.3, it is shown the plot of the cosmographic Amati re-lation. The confidence level curves are calculated as the 3 σ de-viation from the best fit. Note how the 13 GRBs added to the oldsample, marked with a circle, are distributed about the best-fitcurve, indicating that the spectral analysis of these 13 GRBs iscorrect.
5. Discussion and Conclusions
The issue to extend the cosmic scale ladder up to medium-highredshift is an important questions of modern cosmology. A pos-sible way to achieve this goal is to take into account GRBs, themost powerful explosions in the Universe. The energy emittedby these objects spans about six orders of magnitude. However,they cannot be assumed as standard candles in a proper sense.Dispite of this lack, the existence of several correlations betweenspectroscopic and photometric observable quantities of GRBsallow us to solve in part this problem. The fundamental pre-requisite to obtain such relations is to estimate the emitted en-ergy in a way independent of the cosmological model. In this pa- . Capozziello and L. Izzo: A cosmological independent calibration of the E p , i - E iso correlation for Gamma Ray Bursts 5 OO OOO OO OOOOO O E peak (keV) E iso (ergs) 10 100 100010 Fig. 3.
Plot of the cosmographic Amati relation. The line of prediction bounds represents a deviation of 3 σ from the best fit line, thethick line. The circle represents the 13 GRBs added to the old sample, (Amati et al. 2009). y Μ Fig. 2.
Plot of µ ( y ) computed for a fiducial Λ CDM cosmolog-ical model, the continuous line, and for the reconstructed µ ( y )obtained by the cosmographic fit of the SNeIa, the dashed line.Note the slight deviation at very high redshift.per, we have considered a relation for the luminosity distance d l that is independent on the dynamics of the Universe, but, in prin-ciple, could work only at small redshift. Although we have usea parameterization for the redshift which allows to transform thevariable z in a new variable y , ranging in a limited interval, wehave seen that the obtained luminosity distance at high-redshiftdi ff ers slightly from the fiducial model Λ CDM at high redshift,see Fig.3. Nevertheless, since we obtained the curve d l ( y ) by ananalysis of the SNeIa Union survey, that extends up to a red-shift of ∼ . By the way using the d l ( y ) obtained with the Estimates of the Baryonic Acoustic Oscillations (BAO) performedby forthcoming surveys of clusters at intermediate redshift ( z ≈ d l ( y ). cosmographic fit of the SNeIa, we have constrained a sample ofGRBs in a cosmology-independent way so that we have fitteda cosmographic Amati relation for GRBs. The results are simi-lar to those obtained from other analysis performed using othermethods, (Schaefer 2006; Liang et al. 2008; Amati 2002). It isimportant to stress the independence from cosmology and thecalibration obtained by SNeIa. In our opinion, this characteristicis relevant, from one side, to constrain cosmological models, inparticular, dark energy models, and, from another side, to checkthe physical validity of the Amati relation. Acknowledgements.
We warmly thank L. Amati for providingus the GRB sample in (Amati et al. 2009). LI warmly thanks alsoR. Benini for useful discussion and help with the Mathematicapackage data analysis.
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