Cosmology using Long Gamma Ray Bursts: Statistical Analysis of Errors in the calibrated Data

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1 Research in Astron. Astrophys Vol. X No. XX, Research in Astronomy and Astrophysics Cosmology using Long Gamma Ray Bursts: Statistical Analysis of Errors in the calibrated Data Meghendra Singh 1, Shashikant Gupta 2, Amit Sharma 3 and Satendra Sharma 4 1 Dr.A.P.J.Abdul Kalam Technical University, Lucknow, , Uttar Pradesh, India. meghendrasingh_db@yahoo.co.in 2 G.D Goenka University Gurgaon, Haryana , India. 3 Amity University Haryana, Gurgaon (Manesar), Haryana , India. 4 Yobe State University, Damaturu, Yobe State, Nigeria. Received 2018 March 18; accepted 2018 August 29 Abstract We investigate direction dependence and non-gaussian features in the High-z cosmological data using the statistics and Kolmogrov Smirniov test. These techniques are applied on a set of calibrated Long Gamma Ray Bursts (GRBs) and its combination with the latest Type Ia Supernovae data (Union2). Our statistical analysis shows a weak but consistent direction dependence in both the data sets. The analysis also indicates non- Gaussian nature of errors in both data sets. Key words: gamma rays bursts, observational cosmology, cosmological parameters, Gaussian distribution, supernovae. 1 INTRODUCTION Gamma-ray bursts (GRBs) are among the most energetic explosion in the Universe. Their high energy photons in the gamma-ray band are almost unaffected from dust extinction, as a result they are detectable up to very high redshifts (Salvaterra R. et al. 2009; Cucchiara A. et al. 2011). Long GRBs (for which pulse duration is greater than 2s) can be used to investigate the Universe at high redshifts which is difficult to access by Type Ia Supernovae (SNe Ia); thus long GRBs have been proposed as a complementary probe to SNe Ia. However, due to an extensive variety of isotropic equivalent luminosities and energy outputs (Ghirlanda 2006) GRBs can not be treated as standard candles. Efforts have been made to calibrate long GRBs using the following empirical relations: Isotropic equivalent radiated energy (E iso ) and peak energy (E p ) correlation for GRBs (Amati 2008). Peak energy (E p ) in a vfv spectra and collimated emission (E γ ) correlation for GRBs (Ghirlanda 2004). Unfortunately, the physical interpretation of the above correlations is not well understood. Thus, standardization of these correlations is subject to availability of low redshift GRBs. Since not many low redshift GRBs are available (Kodama 2008);(Li 2008), the calibration is obtained by assuming a specific cosmology (specific values of cosmological parameters). As a result the GRB data obtained from this calibration depends on cosmology. Hence, it can not be further used for estimation of cosmological parameters. This problem is known as "circularity problem" in GRB Cosmology. It should be noted that while deriving distances to SNe Ia sometimes numerical value of Hubble constant is inserted by hand

2 2 M.Singh et al. (Amanullah et al. 2010) which makes it cosmology-dependent. However, one can easily overcome this problem by making use of nearby SNe Ia. In order to use GRBs as cosmological probes one needs specific statistical techniques to avoid circularity problem. Many cosmology-independent methods have been proposed to achieve this, e.g., collimation-corrected energy method (Ghirlanda 2004), the luminosity indicator method (Liang 2005) derived from 157 SNe Ia, the Bayesian method (Firmani 2005), Markov Chain Monte Carlo (MCMC) method (Li 2008). Recently (Jing Liu & Hao Wei 2015) have used Padé approximation to calibrate high-redshift GRBs which can be used to constrain cosmological models. Cosmological principle(hereafter CP) has a significant role in modern cosmology; it states that the Universe is homogeneous and isotropic on large scales (Weinberg 2008). However, it is essential and important to check if the cosmological principle is consistent with the latest observational data. As more high-redshift GRB data is now available (Liang 2008; Hao Wei 2010), it becomes possible to search for any possible anisotropy in the high redshift universe. Earlier Gupta, Saini & Laskar (2008), Gupta, Saini (2010) and Gupta, Singh (2014) (hereafter GS14) have investigated the direction dependence using the techniques based on extreme value statistics through different sets of SNe Ia. As a biproduct of the technique they also obtained information about non-gaussian features in the data. In the present work we investigate the direction dependence and non-gaussianity in the latest long GRB data. The paper is structured in the following way. The data analyzed for the current work is briefly discussed in Section 2, while the Methodology, i.e., the χ 2 statistic, χ statistic and KS Test is explained in Section 3. Results and conclusions have been presented in Section 4 and Section 5 respectively. 2 DATA SET In this paper we consider a sample of 79 calibrated long GRBs from (Jing Liu & Hao Wei 2015) at high-redshift (z > 1.4) which is claimed to be free from the circularity problem and thus can be used to test the cosmological models. For our analysis we use Union2 data from (Amanullah et al. 2010), which contain 557 SNe Ia with redshift up to (z = 1.4) and their combination (hereafter UNGRB) having636 data points. 3 METHODOLOGY 3.1 The χ 2 statistic For a given GRB, the difference between apparent magnitude (m(z)) and absolute magnitude (M) is called distance modulus (µ), µ(z) = m(z) M, (1) where m(z) depends on the intrinsic luminosity of GRB, z is the redshift of GRB. We can also express distance modulus (measured in Mpc) as, For ΛCDM universe D L can be expressed as, µ(z) = 5log(D L (z))+25, (2) D L (z) = c(1+z) H 0 z 0 dx h(x), (3) whereh(z;ω M,Ω X ) = H(z;Ω M,Ω X )/H 0, and thus depends only on the cosmological parameters matter densityω M and the dark energy densityω X. For a flatλcdm universe we calculate bestfit values for complete data set byχ 2 minimization, which we expressed as, χ 2 = Σ N i=1[ (µ i obs µ i th)/σ i ] 2, (4)

3 Cosmology Using Long Gamma Ray Bursts:... 3 where total number of data points given by N. (µ i obs ) is the measured distance modulus from the data, andµ i th is the calculated theoretical distance modulus. Observed standard error inµi is denoted by σ i. We obtain χ i for each GRB as, χ i = [µ i obs µ i th(z i ;Ω M )]/σ i. (5) Whereµ i th (z i;ω M ) is calculated by using the best fit values ofω M andh 0. We divide the data into two hemispheres labeled by the direction vector ˆn, and take the difference of the χ 2 computed for the two hemispheres separately ( χ2ˆn ). Now to calculate the maximum absolute difference we rotate the direction vector ˆn and calculate the χ 2 several times (for further details see GS14). χ 2 = max{ χ 2ˆn }. (6) We generate the distribution of χ 2 by shuffling the data valuesz i,µ i andσ i over the SNe positions and refer to it as bootstrap distribution. The distribution when plotted looks skewed as expected from extreme value theory (Haan L. & Ferreira A. 2006). The greater distance of original χ 2 (without shuffling the data values) from the peak of bootstrap distribution would indicate that the SNe positions in the data are important i.e., it indicate direction dependence in the data. Since, analytic expression for the distribution of χ 2 is not available one can obtain the theoretical distribution numerically and compare it with the bootstrap distribution. If the errors (σ i s) in the data follow the normal distribution than the χ i s in Eq. 5 would obey standard normal distribution. To obtain the theoretical distribution one can generateχ i s using standard normal distribution instead of Eq. 5. If the bootstrap and theoretical distribution do not match it would indicate that χ i s in Eq. 5 do not follow standard normal distribution and hence the errors (σ i s) have the non-gaussian nature. 3.2 The Delta-Chi( χ) Statistic χ i defined in Eq. 5 could be positive or negative, but the information is lost when we square it. Thus χ 2 statistic does not contain information about the SN being above or below the fit. We consider another method without squaring the χ i s that does contain information regarding whether the SN at a redshift is closer or farther from us. An additional advantage of this method is that, the distribution of maxima can be calculated analytically as discussed below. We have two subsets of data represented by two hemispheres and labeled by the direction vector ˆn, have N north and N south supernovae, where N is the total number of SNe, N = N north +N south, and define the quantity χˆn = 1 N N north i=1 N south χ i j=1 χ j. (7) From Eq. 7 it is clear that χˆn = 0 and ( χˆn ) 2 = 1. It follows from the Central Limit Theorem that if N 1, the quantity χˆn obeys a normal distribution with a zero mean and unit standard deviation. To obtain the maximum absolute difference we maximize χˆn by varying the direction ˆn across the sky. χ = max{ χˆn }. (8) Extreme value theory shows that the distribution of χ follows a two parameter distribution known as Gumbel distribution and is given by (Haan L. & Ferreira A. 2006); P( ) = 1 [ s exp m ] [ ( exp exp m )] s s, (9)

4 4 M.Singh et al. Where the position parameter m and the shape parameters s can be determined analytically in the theoretical limit of χ statistic. In the limit (N dir 1, wheren dir is number of independent directions) the parameters are given by; m = 2logN dir loglogn dir log4π ;s = 1 m. (10) Here it is assumed that the total number of SNe is large, i.e.,n 1, since the distribution for χˆn becomes Gaussian only in this limit. 3.3 Kolmogrov Smirniov Test Finally we apply the Kolmogrov Smirniov Test (KS Test) which is described below. Originally it was presented in (Singh et al. 2016) and in (singh et al. 2016) to investigate non-gaussianity in the HST Key Project Data and in SNe data. Ifµ i th (z) is the theoretical value of the distance modulus forith GRB at redshiftz then the observed value µ i obs will be µ i obs = µ i th(z)±σ i ; (11) As discussed earlier, χ i should follow standard Gaussian distribution, to check this, we use KS test (Press 2007). We define our null hypothesis as: "The errors in GRB data are drawn from a Gaussian distribution". If null hypothesis is true χ i s in Eq. 5 would follow standard Gaussian distribution. To test this we use Matlab function kstest[h,p,k,cv]; where: p is the probability of the data errors being drawn from Gaussian distribution, k represents the maximum distance between the cumulative distribution (CDF) of χ i s with that of standard Gaussian. cv is the critical value which is decided by the significance level (α). Different values of α, indicate different tolerance levels for false rejection of the null hypothesis.cv is the critical value of the probability to obtain the data set in question given the null hypothesis; and is to be compared with p. A value h = 1 is returned by the test if p < cv and the null hypothesis is rejected. On the other hand ifp > cv,hremains 0 and the null hypothesis is not rejected. 4 RESULTS First we calculate the best fit values of matter density (Ω M ) and Hubble constant (H 0 ) for both GRB and SNe data. It is clear from the Table 1 that GRB data favor a smaller value ofω M and larger value of H 0 compared to Union2. Smaller value of χ 2 in case of GRBs indicates that the errors may have been overestimated. The bestfit values of Ω M and H 0 are same for Union2 and UNGRB data, expressing the fact that SNe dominate in the combined data set. The direction of maximum discrepancy and the value of ( χ 2) and ( χ ) are shown in Table 2 and Table 3. The values ofp,k andcv for various data sets are presented in Table 4. For reference the results of simulations for each data set are plotted in part of Figure 1, 2, 3, 4, and 5. A specific bias related to the bootstrap distribution was discussed in GS14. The theoretical distribution is obtained by assuming that the χ i s follow normal distribution with zero mean and unit standard deviation, thus theoretical χ i s are unbounded. On the other hand, the bootstrap distribution is prepared by shuffling through a specific realization of χ i, which have a maximum value for some SN/GRB. Thus the χ i s in bootstrap distribution have an upper bound due to which the bootstrap distribution will be shifted slightly to the left of the theoretical distribution. It can also be seen in the simulation figure of Figure 1, 2, 3, 4, and 5.

5 Cosmology Using Long Gamma Ray Bursts:... 5 Fig. 1: Simulation figure for 79 GRBs, shows comparison of bootstrap and theoretical probability distributions. Positions of GRBs were generated randomly on the sky. The bootstrap and theoretical probability distributions of 79 GRBs for the χ 2 statistic. Shape of the bootstrap distribution is different from that in Figure 1. The shifting of bootstrap on left indicates non-gaussianity. Table 1: Bestfit values of cosmological parameters (ΛCDM) for different data sets. Set Data Ω M H 0 χ 2 /dof GRB Union UNGRB Table 2: Direction of maximum χ 2 for different data sets. long and lat stand for longitude and latitude respectively in Galactic Coordinate System. Set Data χ 2 long lat GRB Union UNGRB Results for ( χ 2) Statistic GRBs: The theoretical and bootstrap distributions for the set of 79 GRBs are shown in part of Figure 1. Comparison with the Figure 1 exhibits that the shift in bootstrap is much larger than expected, suggesting the non-gaussian nature of errors. The location of χ 2 for GRBs is nearly 2σ away from the maxima of the bootstrap distribution which suggests directional dependence in the data. Union2: The results of ( χ 2) statistic for Union2 data have been discussed in GS14. The bootstrap and theoretical distributions have been shown in Figure.5 of GS14.

6 6 M.Singh et al. Fig. 2: Simulation figure for UNGRB, shows comparison of bootstrap and theoretical probability distributions. The bootstrap and theoretical probability distributions of UNGRB for the χ 2 statistic. In comparison with Figure 2 bootstrap distribution appears on the right side which shows a sign of non-gaussianity. Table 3: Direction of maximum χ for different data sets. long and lat stand for longitude and latitude respectively in Galactic Coordinate System. Set Data χ long lat GRB Union UNGRB Table 4: Results of KS test for different data sets at 1% significance level (α). Set p value k Cv GRB Union UNGRB UNGRB: Fig 2 represents the two distributions for UNGRB. Comparison with Fig 2 shows that the leftward shift of bootstrap distribution is much smaller than expected, indicating slight non-gaussianity for the residuals. χ 2 for UNGRB is about one sigma away from the peak which indicates weak directional dependence in UNGRB data. It should be noted that the direction dependence is mild as compared to GRB data.

7 Cosmology Using Long Gamma Ray Bursts:... 7 Fig. 3: Simulation figure for 79 GRBs, shows comparison of bootstrap and theoretical probability distributions. The bootstrap, theoretical and analytic probability distributions for the χ statistic for GRBs. The analytic distribution obtained from the Eq. 9 with parameters defined in Eq. 10.The bootstrap distribution shows anisotropy at one sigma level. Fig. 4: Simulation figure for 557 SNe, shows comparison of bootstrap and theoretical probability distributions. The bootstrap, theoretical and analytic probability distributions for the χ statistic for Union2 data. The bootstrap distribution indicates anisotropy away from the one sigma level.

8 8 M.Singh et al. Fig. 5: Simulation figure for 636 data points, shows comparison of bootstrap and theoretical probability distributions. The bootstrap, theoretical and analytic probability distributions for the χ statistic for UNGRB data. The bootstrap distribution indicates anisotropy at two sigma level. 4.2 Results for ( χ ) Statistic GRB: χ statistic has an advantage, since the distribution can be calculated analytically. The theoretical, bootstrap and analytic distributions for the set of 79 GRBs are plotted in Fig 3. The χ for GRBs is more than 1σ away from the mode of bootstrap distribution which shows slight direction dependence in GRBs. The analytic distribution is obtained using Eq. 9 with parameters defined in Eq. 10. The shape of analytic distribution is quite different. The mismatch could arise if all the directions are not independent. In this case the number of actually independent directions would be less thann d and hence actual value ofmwould be smaller than calculated in Eq. 10. This also makes the spread larger in bootstrap & theoretical distribution compared to the analytic one. Union2: Theoretical, bootstrap and analytic distributions for Union2 are plotted in Fig 4. Since the theoretical χ i s are unbounded while the bootstrap χ i s are bounded, the bootstrap distribution should lay slightly on the left of the theoretical distribution. As shown in Fig. 4 in contrast to the above expectation, Fig. 4 shows that the bootstrap distribution match quite well with the theoretical one, indicating slight non-gaussian features in the errors. UNGRB: The theoretical, bootstrap and analytic distributions for UNGRB data have been plotted in Fig 5. Although the analytic and bootstrap distributions appear quite different in the figure, the bootstrap and theoretical distributions match quite well. Comparison with Fig 5 indicate weak signature of non-gaussianity. χ is nearly 2σ away from the mode of the bootstrap distribution. 4.3 Results for KS Test We calculateχ i s as defined in Eq 5 for GRB data using the best-fit values presented in Table 1. Further, we generate a set of 79 random numbers following Gaussian distribution with zero mean and unit standard deviation which represent the theoretical χ i s. Fig. 6 shows the histogram of both the χ i s. It is

9 Cosmology Using Long Gamma Ray Bursts: GRB Standard Normal Fig. 6: Histogram of χ i for GRB and UNGRB data is compared with that of standard normal distribution. clear from the figure that the spread in χ i s from GRB data is smaller compared to the theoretical χ i s. Fig. 6 shows a similar graph for the UNGRB data comprising of 636 data points. Fig. 7 along with Table 4 represents the main result of KS test. Fig. 7 shows the cumulative distribution of χ i s from GRB and theoretical χ i s. While Fig. 7 represents the same in case of UNGRB. The second, third and fourth column in Table 4 denote values of p, k and cv respectively. Since p > cv in all cases giving h = 0, which means that we can not reject the null hypothesis that the errors have been drawn from a Gaussian distribution. However, Fig. 7 shows CDF of GRB χ i s has large deviation from the standard normal compared to the UNGRB counterpart. 5 CONCLUSIONS In this paper we have presented our result for the GRB/UNGRB data using the statistics discussed in Section 3.1, Section 3.2 and Section 3.3. As discussed earlier χ 2 and χ statistic are based on extreme value theory. The Later has advantage over the former since its distribution can be calculated analytically. Both χ 2 and χ quantify the direction dependence in the data; they also qualitatively indicate the presence/absence of non-gaussian features in the data. The KS Test (Section 3.3) on the other hand quantitatively measures the non-gaussianity in the data. Our main conclusions for this piece of work are: Set of79 GRBs show evidence for non-gaussian behavior of errors, UNGRB data also shows non- Gaussian nature of errors but weaker compared to GRBs. Set of 79 GRBs is different from the peak of the bootstrap distribution about 2σ; while UNGRB is different from the peak by about 1σ. It shows a weak but consistent direction dependence in the both data sets. This consistency indicates a preferred direction for both data sets. The maximum anisotropy from different measurements, are presented in Table 2 and Table 3. The directions for Union2 and UNGRB are similar for χ statistic. In all cases we see that results of UNGRB are very similar to Union2. It could be due to the fact that the SNe are greater in number and hence dominate in the UNGRB data.

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