| BZs >gnuplot.ps Acta Polytechnica Vol. 51 No. 2/2011 Cosmology with Gamma-Ray Bursts Using k-correction A. Kovács, Z. Bagoly, L. G. Balázs, I. Horváth, P. Veres Abstract In the case ofGamma-RayBursts withmeasured redshift, we can calculate the k-correction to get the fluence and energy thatwere actually produced in the comoving system of theGRB. To achieve this we have to usewell-fitted parameters of a GRB spectrum, available in the GCN database. The output of the calculations is the comoving isotropic energy Eiso, but this is not the endpoint: this data can be useful for estimating the ΩM parameter of the Universe and for making a GRB Hubble diagram using Amati’s relation. Keywords: k-correction, Gamma-Ray Burst, cosmology, Hubble diagram, density parameter of matter. 1 Introduction Several papers present how to make k-corrections (e.g. [1]). We will only note here the principles and the most important considerations. A typical GRB spectrum has three main available parameters: peak energy (Epeak ), low- and high energy spectral indices (α, β). With the Band function [2] and the Cutoff power-law function (hereafter CPL) these are well- fitted. The motivation of k-correction is related to the fact that the energy distribution measured with gamma-ray detectors placed on satellites is not equal to the energy that the burst released in its comoving system. The goal is to correct the redshifted GRB spectrum. We will now show the main steps of this method and the meaning of k. 2 Theory of the k-correction By definition, fluence is the integral of the Band/CPL function over the energy range in which the detector is responsive (see equation 1). If redshift is taken into account, the formula for bolometric energy has to be modified. S[a,b] = ∫ b a Φ(E) dE (1) E[E1,E2] = 4πD2L (1 + z) S[Emin,Emax] · (2) S[E1/(1+z),E2/(1+z)] S[Emin,Emax] where E1 = 1 keV and E2 = 10 000 keV, conven- tionally. The detectors do not measure the fluence between [E1/(1 + z), E2/(1 + z)]. All of them have an [Emin, Emax] interval that they detect in, so the fluence should be corrected, as can be seen in equa- tions (2) and (3). Hereafter E[E1,E2] is named Eiso. E[E1,E2] = 4πD2L (1 + z) S[Emin,Emax] · k (3) Now we can see the meaning of k: it is a fac- tor that multiplies the fluence, and therefore the en- ergy. The most necessary parameters are also clear. The Gamma-Ray Burst Coordinate Network (GCN) provides a good database for the calculations. We only have to search for GRBs with measured redshift and decide whether the fit of the spectrum contains a peak energy parameter or not. This is important, because it is needed when working with Amati’s re- lation [3]. We finally found 72 GRB samples, most of which are Konus-WIND data. However, some data is from other sources. Of course, the data of one GRB sample is not mixed. It is important to say some words about the error calculations. We used a somewhat unusual method: because each measured parameter in the GCN database has an error, we generated random Gaussian distributions where the mean value was the measured number, and the vari- ance was the error. The k-corrections were made with these numbers from the tails of the curves, so we fi- nally obtained a distribution for every GRB energy that has a mean value and a variance. We thus iden- tified the error as the variance in each case. With this procedure the ΔEiso errors are approximately one order less than the Eiso energies. 3 Results of the k-corrections We can clearly see from Figure 1 that the most fre- quented k value is approximately 1.2. Figure 2 shows the distribution of the isotropic energies. These en- ergies are used when we want to test the Amati re- lation [4], which states that there is a correlation be- tween Epeak (1 + z) (Epeak in keV) and log10(Eiso) (see Figure 3). There are some outlier points. These are short GRBs (def.: t90 < 2s) which do not fol- low this relation. The stars refer to points using the CPL function, the squares are from the Band func- tion. Our group wanted to test the trimodality of the GRB distribution. As mentioned above, short GRBs 68 Acta Polytechnica Vol. 51 No. 2/2011 0.5 1.0 1.5 2.0 2.5 3.0 k 0 10 20 30 40 N Fig. 1: Distribution of k 50 51 52 53 54 55 log 10 E iso 0 5 10 15 20 N Fig. 2: Distribution of log10(Eiso), Eiso is measured in ergs 50 51 52 53 54 55 log 10 E iso 10 100 1000 10000 E p e a k( 1 + z) Fig. 3: Visualization of Amati’s relation do not follow the Amati’s relation, but the others can also be separated into groups in the Amati plane. Figure 4 shows these datapoints. There is an overlap between the intermediate (2s < t90 < 10s) and the long type groups (signed ∗), so the separation cannot be seen in our data. With more burst in the future it may be visible. 50 51 52 53 54 55 log 10 E iso 10 100 1000 10000 E p e a k( 1 + z) Fig. 4: The intermediate and the long GRB groups 4 Cosmological applications We will now present the most interesting results that can be derived form the dataset, the Hubble dia- gram and an estimation of ΩM . The first interest- ing result is based on two important things: accel- erating expansion of the Universe from Supernova Cosmology Project data [5] and earlier results us- ing GRBs [6]. Our method was very simple, as we just wanted to make estimations. We were curious whether the time-dilatation effect is seen or not. Let us note the steps in this procedure: first of all, con- sider that the Amati relation is real and fit a straight line to the datapoints. After this, we can calculate DL using the Eiso values related to the two param- eters of the straight line. Finally, we ought to think that the points are exactly on the line. In this way we can get the luminosity distance in a different way, as written below in equation (4). DL = ( Eiso(1 + z) S[Emin,Emax]4πk )1 2 (4) The last step in this operation is to put the de- rived DL to the well-known distance modulus formula (see equation 5), and the final result is Figure 5. 0 2 4 6 8 10 z 40 45 50 55 μ( m a g ) Fig. 5: The GRB Hubble diagram 69 Acta Polytechnica Vol. 51 No. 2/2011 0.0 0.2 0.4 0.6 0.8 1.0 Ω Μ 1.440 1.442 1.444 1.446 1.448 1.450 1.452 1/ ρ Fig. 6: A cosmological probe μ = 5 log10 DL − 5 (5) It is seen that the points are mostly above the the- oretical curve, so it is clear time-dilatation causes the same effect as in the case of supernovae. The fi- nal result is the most important one: an estimation of ΩM , the density parameter of matter. When the k-corrections were calculated, we had to give some input parameters, for example the mentioned quan- tity, because DL depends on the cosmology (see equa- tion 6). It is clear that if we change these parameters, Amati’s relation will also change. DL(z, Ωm, ΩΛ, H0) = c(1 + z) H0 · (6) 1∫ z 0 [(1 + z′2)(1 + Ωmz′) − z′(2 + z′)ΩΛ] 1 2 dz′ We can ask with what conditions we can get the highest correlation on the Amati plane. To find the answer, the calculations have been redone with so many values of ΩM , considering that Ωtotal = 1, i.e. a flat Universe, but there is a chance to test other mod- els, too. We wanted to represent the correlation sim- ply with the correlation coefficient ρ (see Figure 6). This has a maximum when the correlation is highest, but all the results of some earlier papers [7] use meth- ods where there is a minimum in the same case, and we therefore used the 1/ρ form in Figure 6. We can see that the minimum is at 0.2, which would be the optimal ΩM for obtaining the highest correlation. 5 Conclusions Our method, which is based on the k-correction, is a useful tool for cosmology. However, our estima- tion of ΩM is less than the obtainable value from the WMAP data [8], which may be the most precise data that is available. Our results are approximately the same as the results of earlier GRB studies, although we have used different data. Acknowledgement The Project is supported by the European Union, co- financed by the European Social Fund (grant agree- ment no. TAMOP 4.2.1./B-09/1/KMR-2010-0003), and in part through OTKA K077795, OTKA/NKTH A08-77719, and A08-77815 (Z.B.) grant. References [1] Bloom, J. et al.: The Prompt Energy Re- lease of Gamma-Ray Bursts using a Cosmological k-Correction, The Astronomical Journal, 2001, Vol. 121, Issue 6, pp. 2 879–2 888. [2] Band, D. et al.: BATSE observations of gamma- ray burst spectra. I – Spectral diversity, The As- trophysical Journal, Vol. 413, pp. 281–292. [3] Amati, L. et al.: Intrinsic spectra and energet- ics of BeppoSAX Gamma-Ray Bursts with known redshifts, Astronomy and Astrophysics, Vol. 390, pp. 81–89. [4] Amati, L.: The Ep,i- Eiso correlation in gamma- ray bursts: updated observational status, re- analysis and main implications. Monthly No- tices of the Royal Astronomical Society, 2006, Vol. 372, Issue 6, pp. 233–245. [5] Perlmutter, S. et al.: Measurements of Omega and Lambda from 42 High-Redshift Supernovae, The Astrophysical Journal, Vol. 517, Issue 2, pp. 565–586. [6] Schaefer, B. E.: The Hubble Diagram to Redshift > 6 from 69 Gamma-Ray Bursts, The Astrophys- ical Journal, 2007, Vol. 660, Issue 1, pp. 16–46. [7] Amati, L. et al.: Measuring the cosmologi- cal parameters with the Ep,i-Eiso correlation of gamma-ray bursts. Monthly Notices of the Royal Astronomical Society, 2008, Vol. 391, Issue 2, pp. 577–584. [8] Komatsu, E. et al.: Five-Year Wilkinson Mi- crowave Anisotropy Probe Observations: Cosmo- logical Interpretation, The Astrophysical Journal Supplement, Vol. 180, Issue 2, pp. 330–376. András Kovács Dept. of Physics of Complex Systems Eötvös University H-1117 Budapest, Pázmány P. s. 1/A, Hungary 70 Acta Polytechnica Vol. 51 No. 2/2011 Zsolt Bagoly Dept. of Physics of Complex Systems Eötvös University H-1117 Budapest, Pázmány P. s. 1/A, Hungary Lajos G. Balázs Konkoly Observatory H-1525 Budapest, POB 67, Hungary István Horváth Dept. of Physics Bolyai Military University Budapest, POB 15, H-1581, Hungary Péter Veres Dept. of Physics of Complex Systems Eötvös University H-1117 Budapest, Pázmány P. s. 1/A, Hungary Dept. of Physics Bolyai Military University Budapest, POB 15, H-1581, Hungary 71