The truncated lognormal distribution as a luminosity function for SWIFT-BAT gamma-ray bursts

The determination of the luminosity function (LF) in gamma ray bursts (GRBs) depends on the adopted cosmology, each one characterized by its corresponding luminosity distance. Here we analyse three cosmologies: the standard cosmology, the plasma cosmology, and the pseudo-Euclidean universe. The LF of the GRBs is firstly modeled by the lognormal distribution and the four broken power law, and secondly by a truncated lognormal distribution. The truncated lognormal distribution fits acceptably the range in luminosity of GRBs as a function of the redshift.

cosmology is the ΛCDM cosmology, but other cosmologies such as the plasma or the pseudo-Euclidean 20 cosmology can also be analysed. Once the luminosity is obtained, we can model the LF by adopting the 21 lognormal distribution, see [2,3] and by a four broken power law. 22 In the hypothesis that the luminosity of a GRB is due to the early phase of a supernova (SN), the 23 minimum and maximum are due to the various parameters which drive the SN's light curve, see [4]. In the framework of the standard cosmology, the received flux, f , is where D L (z) is the luminosity distance, which depends on the parameters of the adopted cosmological model and z is the redshift. As a consequence, the luminosity is The above formula is then corrected by a k-correction, k(z, γ), where k(z, γ) = 10 4 keV 1keV where C is a constant and γ is the observed spectral index in energy, see [5] for more details. The corrected luminosity is therefore L = 4πD L (z) 2 f k(z, γ) .
In the case of the survey from the 70 month SWIFT-BAT, the flux f is given in f W m 2 , γ and z are positive 31 numbers, see [1]; Table 1 reports a test-GRB.  A further application of the minimax rational approximation, which is characterized by the two parameters p and q, allows finding a simplified expression for the luminosity distance, see eqns (33a) and (33b) in [6]. The above minimax approximation when p = 3, q = 2 is D L,3,2 = p 0 + p 1 z + p 2 z 2 + p 3 z 3 q 0 + q 1 z + q 2 z 2 M pc , and Table 3 reports the coefficients for the two compilations here used. The monochromatic luminosity, X-band (14-195 keV), without k −z correction, log(L 3,2 ) b according to eqn (2) is log(L 3,2 (erg s −1 )) b = 0.43429 ln 1.1964 fluxfwm2 ( In the case of a test-GRB with parameters as in Table 1, the above formula gives log(L) = 48.13 against log(L SW IF T ) = 48.01 of the SWIFT-BAT catalog. The goodness of the approximation is evaluated through the percentage error, η, which is

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and over all the elements of the SWIFT-BAT catalog 2.28 10 −5 % ≤ η ≤ 0.295%. We now report an expression for the luminosity of a GRB, eqn (4), based on the minimax approximation when the Union 2.1 compilation is considered log(L 3,2 (erg s −1 )) = 41.5647+

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In the case of a test-GRB with parameters as in Table 1, the above formula gives log(L) = 54.512 which means a bigger luminosity of ≈ 6 decades with respect to the band luminosity. Figure 2 reports the luminosity-redshift distribution for the SWIFT-BAT survey as well a a theoretical lower curve which can be found by inserting the minimum flux in eqn (8). Another useful quantity is the angular diameter distance, D A , which is see [7], and therefore D A,3,2 = D L,3,2 (1 + z) 2 . (10) 2.3. Luminosity in the plasma cosmology 40 The distance d in the plasma cosmology has the following dependence: see [8][9][10][11] and Table 4. The monochromatic luminosity, X-band (14-195 keV), is Version October 1, 2018 submitted to Galaxies In the case of a test-GRB with parameters as in Table 1, the above formula gives log(L) = 46.63, which 41 a lower value than the log(L SW IF T ) = 48.01 of the SWIFT-BAT catalog.

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The luminosity in the case of the absence of absorption is where the k(γ) correction is There is no relativistic correction in the denominator because the plasma cosmology is both static and 43 Euclidean. Figure 3 reports the luminosity in the plasma cosmology as a function of the redshift as well 44 as the theoretical luminosity. The distance d in the pseudo-Euclidean cosmology has the following dependence: and we used H 0 = 67.93km s −1 Mpc −1 , see Table 5. The above formula gives approximate results up to z 1.0. The monochromatic luminosity, X-band (14-195 keV), is where the k(z) correction is absent or 2.5. High versus low z 48 The differences between the four distances here used, which are the luminosity distance and the angular-diameter distance in the ΛCDM, the plasma cosmology distance, and the pseudo-Euclidean cosmology distance, can be outlined in terms of a percentage difference, ∆. As an example for D A , Figure 4 reports the four distances and for z ≤ 0.05 the three percentage differences are lower than 49 10%. In the framework of the two Euclidean distances, the plasma and the pseudo-Euclidean one, for The four broken power law has the following piecewise dependence: each of the four zones being characterized by a different exponent α i . In order to have a PDF normalized to unity, one must have For example, we start with c 1 =1: c 2 will be determined by the following equation: where is a small number, e.g. = L 2 10 +8 . This PDF is characterized by 9 parameters and takes values L 57 in the interval [L 1 , L 5 ].

Lognormal distribution 59
Let L be a random variable taking values L in the interval [0, ∞]; the lognormal probability density function (PDF), following [12] or formula (14.2) in [13], is where L * = exp µ LN and µ LN = ln L * . The mean luminosity is The distribution function (DF) is where erf(z) is the error function, see [14]. A luminosity function for GRB, P DF GRB , can be obtained by multiplying the lognormal PDF by Φ * , which is the number of GRB per unit volume, Mpc 3 units for unit time, yr units, A numerical value for the constant Φ * can be obtained by dividing the number of GRB, N GRB , observed in a time, T , in a given volume V by the volume itself and by T , which is the time over which the phenomena are observed, in the case of SWIFT-BAT, 70 month, see [1], where the volume is different in the three cosmologies, where D A,3,2 has been defined in eqn (10). The parameters of the fit for the four broken power law's 60 PDF are reported in Table 6 when the luminosity is taken with the k(z) correction, Figure 5.

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The parameters of the fit for the lognormal PDF are reported in Table 7 when the luminosity is taken 62 with the k(z) correction.

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The case of LF modeled by a lognormal PDF with L as represented by a monochromatic luminosity 64 in the X-band (14-195 keV) is reported in Table 8.
The goodness of the fit with the lognormal PDF has been assessed by applying the Figure 5. Observed DF (step-diagram) for GRB luminosity and superposition of the four broken power laws' DFs (line), case of ΛCDM cosmology with parameters as in Table 6.   Table 7 for the results.

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In the case of the ΛCDM cosmology Figure 6 reports the lognormal DF, with parameters as in Table   71 7.

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In the case of the ΛCDM cosmology, Figure 7 reports a comparison between the empirical distribution 73 and the lognormal PDF, and Figure 6 reports the lognormal DF, with parameters as in Table 7.

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The case of the plasma and pseudo-Euclidean cosmologies are covered in Figs 8 and 9 respectively.   Table  7.  Table 7.  Table 7. We assume that the flux, f , scales as f = L 4πr 2 , according to eqn (15): and The relation between the two differentials dr and dz is The joint distribution in z and f for the number of galaxies is where δ is the Dirac delta function. We now introduce the critical value of z, z crit , which is (33)  Table 7.
The evaluation of the integral over luminosities and distances gives where dΩ, dz and df represent the differential of the solid angle, the redshift, and the flux, respectively, and Φ * is the normalization of the lognormal LF for GRB. The number of GRBs in z and f as given by the above formula has a maximum at z = z pos−max , where which can be re-expressed as (36) Figure 10 reports the observed and theoretical number of GRBs with a given flux as a function of the redshift. The theoretical maximum as given by eqn (35) is at z = 0.017, with the parameters as in Table  7, against the observed z = 0.019. The theoretical mean redshift of GRBs with flux f can be deduced from eqn (34): The above integral does not have an analytical expression, and should be numerically evaluated. The

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A larger number of available GRBs will presumably increase the goodness of the fit. Figure 11. Frequencies of GRBs at a given flux as a function of the redshift, parameters as in Figure 10. The full red line is the theoretical curve generated by dN dΩdzdf (z) as given by the application of the lognormal LF which is eqn (41) in the plasma cosmology with parameters as in Table 8, χ 2 = 6193. 82 We assume that f = L 4πr 2 and

The non-linear case
where r is the distance; in our case, d is as represented by the non-linear eqn (11). The relation between dr and dz is The joint distribution in z and f for the number of galaxies is where δ is the Dirac delta function.

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The evaluations of the integral over luminosities and distances gives dN dΩdzdf = (ln (z + 1)) 2 c 3 √ 2e The above formula has a maximum at z = z pos−max , where where W (x) is the Lambert W function, see [14]. The above maximum can be re-expressed as Figure 11 reports the observed and theoretical number of GRBs with a given flux as a function of the 84 redshift. In the case of the plasma cosmology, the theoretical maximum as given by eqn (42) is at 85 z = 0.0188, with the parameters as in Table 7, against the observed z = 0.019. The theoretical averaged 86 redshift of GRBs is z = 0.041 against the observed z = 0.0385.

The truncated lognormal distribution 88
This section derives the normalization and the mean for a truncated lognormal PDF. This truncated 89 PDF fits the high redshift behaviour of the LF for GRBs. Let X be a random variable taking values x in the interval [x l , x u ]; the truncated lognormal (TL) PDF is Its expected value is The distribution function is The four parameters which characterize the truncated lognormal distribution can be found with the maximum likelihood estimators (MLE) and by the evaluation of the minimum and maximum elements of the sample. The LF for GRB as given by the truncated lognormal, Φ T (L; L * , σ, L l , L u ), is therefore Φ T (L; L * , σ, L l , L u ) = Φ * T L(L; L * , σ, L l , L u ) number M pc 3 yr , where L * is the scale, L l the lower bound in luminosity, L u the upper bound in luminosity and Φ * is 92 given by eqn (27). The LF for GRBs as modeled by a truncated lognormal DF is reported in Figure 12 in the case of the 95 ΛCDM cosmology and in Figure 13 in the case of the plasma cosmology without a k(z) correction; the 96 data is as in Table 9.

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In order to model evolutionary effects, a variable upper bound in luminosity, L u , has been introduced L u = 1.25(1 + z) 2 10 51 erg s , see eqn (7) in [5]; conversely the lower bound, L l was already fixed by eqn (8). A second evolutionary correction is where σ 0 is the evaluation of σ at z ≈ 0, see Table 9.  Table 9. Figure 13. Observed DF (step-diagram) for GRB luminosity and superposition of the truncated lognormal DF (line), the case of the Plasma cosmology without k(z) correction with parameters as in Table 9. Table 9. The 5 parameters of the LF as modeled by the truncated lognormal distribution for z in [0, 0.02] and the two parameters of the K-S test D and P KS . We analysed the case of the ΛCDM cosmology where the luminosity is given by eqn (8) in the second column and the case of the plasma cosmology, the case of the X-band (14-195 keV) without k(z) correction, where the luminosity is given by eqn (12), third column.
Parameter ΛCDM cosmology Plasma cosmology  In the case of the plasma cosmology, the variable upper bound in luminosity, L u , is and Figure 15 reports a comparison between the theoretical average luminosity and the observed average 101 luminosity for the plasma cosmology. ΛCDM without corrections is given as a function of the two observable parameters f luxf wm2 and z, 111 see eqn (6), which can be tested on the SWIFT-BAT catalog of [1].

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Lognormal luminosity function 113 We analysed the widely used lognormal PDF as a LF for GRBs, see Section 3.2. We derived an  The four broken power law PDF gives the best statistical results for the LF of GRBs, see Table 6. The 119 weak point of this LF is in the number of parameters, which is 9, against the 4 of the truncated lognormal 120 LF or 2 of the lognormal LF.

Maximum in flux 122
The maximum in the joint distribution in redshift and energy flux density is here modeled in the case 123 of a pseudo-Euclidean universe adopting a standard technique originally developed for galaxies, see 124 formula (5.132) in [19] and our formula (34). In the case of the plasma cosmology, the maximum has 125 been found by analogy, see our formula (34). In the case of the ΛCDM cosmology, the redshift as a 126 function of the luminosity has a complex behaviour, see formula (66) in [10], and the analysis has been 127 postponed to future research. The above complexity has been considered in an example of a simpler 128 plasma cosmology rather than in the ΛCDM cosmology. see Figure 12 for the case of the ΛCDM cosmology and Figure 15 for the case of the plasma cosmology.

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In other words, the ΛCDM cosmology and the plasma cosmology are indistinguishable in the range of 136 redshifts here analysed, 0 ≤ z ≤ 4.