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Research Article | Open Access
Utilizing the Updated Gamma-Ray Bursts and Type Ia Supernovae to Constrain the Cardassian Expansion Model and Dark Energy
We update gamma-ray burst (GRB) luminosity relations among certain spectral and light-curve features with 139 GRBs. The distance modulus of 82 GRBs at can be calibrated with the sample at by using the cubic spline interpolation method from the Union2.1 Type Ia supernovae (SNe Ia) set. We investigate the joint constraints on the Cardassian expansion model and dark energy with 580 Union2.1 SNe Ia sample and 82 calibrated GRBs’ data . In ΛCDM, we find that adding 82 high- GRBs to 580 SNe Ia significantly improves the constraint on plane. In the Cardassian expansion model, the best fit is and , which is consistent with the ΛCDM cosmology in the confidence region. We also discuss two dark energy models in which the equation of state is parameterized as and , respectively. Based on our analysis, we see that our universe at higher redshift up to is consistent with the concordance model within confidence level.
In recent years, the combined observations of nearby and distant Type Ia supernovae (SNe Ia) have provided strong evidence for the current accelerated expansion of the universe [1–3]. The cause of the acceleration remains unknown. Many authors suggest that the composition of the universe may consist of an extra component called dark energy, which may explain the acceleration of the universe at the current epoch. For example, the dark energy model with a constant equation of state is one of the several possible explanations for the acceleration, while other models suggest that dark energy changes with time, and there are many ways to characterize the time variation of dark energy. Here, we adopt a simple model in which the dark energy equation of state can be parameterized by [4, 5], where is constant, represents the time dependence of dark energy, and is the scale factor. In addition, models where general relativity is modified can also drive universe acceleration, such as the Cardassian expansion model which is a possible alternative for explaining the acceleration of the universe that invokes no vacuum energy .
SNe Ia have been considered a perfect standard candle to measure the geometry and dynamics of the universe. Unfortunately, the farthest SNe Ia detected so far is only at . It is difficult to observe SNe at , even with excellent space-based platforms such as SNAP . And this is quite limiting because much of the most interesting evolution of the universe occurred well before this epoch. Gamma-ray bursts (GRBs) are the most luminous transient events at cosmological distances. Owing to their high luminosities, GRBs can be detected out to very high redshifts . In fact, the farthest burst detected so far is GRB 090423, which is at  (a photometric redshift of 9.4 for GRB 090429B was reported by ). Moreover, in contrast to SNe Ia, gamma-ray photons from GRBs are almost immune to dust extinction, so the observed gamma-ray flux is a direct measurement of the prompt emission energy. Hence, GRBs are potentially more promising standard candles than SNe Ia at higher redshifts. The possible use of GRBs as cosmological probes started to become reality after some empirical luminosity relations were discovered. These GRB luminosity relations have been proposed as distance indicators, such as the correlations , , , [15, 16], , and . Here the time lag () is the time shift between the hard and soft light curves; the luminosity () is the isotropic peak luminosity of a GRB; the variability () of a burst denotes whether its light curve is spiky or smooth, and can be obtained by calculating the normalized variance of an observed light curve around a smoothed version of that light curve ; () is the burst frame peak energy in the GRB spectrum; () is the isotropic equivalent gamma-ray energy; () is the collimation-corrected gamma-ray energy; and the minimum rise time () in the gamma-ray light curve is the shortest time over which the light curve rises by half of the peak flux of the pulse. However,  found that the updated correlation was quite scattered. Its intrinsic scatter has been larger than the one that could be expected of a linear relation.
Generally speaking, with these luminosity indicators, one can make use of them as standard candles for cosmological research. For example,  constructed the first GRB Hubble diagram based on nine GRBs using two GRB luminosity indicators. With the relation,  placed tight constraints on cosmological parameters and dark energy. Reference  used a model-independent multivariable GRB luminosity indicator to constrain cosmological parameters and the transition redshift. Reference  made use of five luminosity indicators calibrated with 69 events by assuming two adopted cosmological models to construct the GRB Hubble diagram. Reference  suggested that the time variation of the dark energy is small or zero up to using the relation. Reference  extended the Hubble diagram up to using 63 gamma-ray bursts (GRBs) via relation and found that these GRB data were consistent with the concordance model within level. In a word, a lot of other works in this so-called GRB cosmology field have been published (please see [19, 25] for reviews). However, there is a so-called circularity problem in the calibration of these luminosity relations. Because of the current poor information on low- GRBs, these luminosity relations necessarily depend on the assumed cosmology. Some authors attempted to circumvent the circularity problem by using a less model-dependent approach, such as the scatter method [26, 27], the luminosity distance method , the Bayesian method [29, 30], and the method by fitting relation parameters of GRBs and cosmological parameters simultaneously [31, 32]. However, these statistical approaches still can not avoid the circularity problem completely, because a particular cosmology model is required in doing the joint fitting. This means that the parameters of the calibrated relations are still coupled to the cosmological parameters derived from a given cosmological model.
To solve the circularity problem completely, one should calibrate the GRB relations in a cosmology-independent way. Recently, a new method to calibrate GRBs in a cosmological model-independent way has been presented [33–35]. This method is very similar to the calibration for SNe Ia by measuring Cepheid variables in the same galaxy, and it is free from the circularity problem. Cepheid variables have been regarded as the first-order standard candles for calibrating SNe Ia which are the secondary standard candles. Similarly, if we regard SNe Ia as the first-order standard candles, we can also calibrate GRBs relations with a large number of SNe Ia since objects at the same redshift should have the same luminosity distance in any cosmology. This method is one of the interpolation procedures which obtain the distance moduli of GRBs in the redshift range of SNe Ia by interpolating from SNe Ia data in the Hubble diagram. Then, if we assume that the GRB luminosity relations do not evolve with redshift, we can extend the calibrated luminosity relations to high-z and derive the distance moduli of high-z GRBs. From these obtained distance moduli, we can constrain the cosmological parameters.
In this paper, we will try to determine the cosmological parameters and dark energy using both the updated 139 GRBs and 580 SNe Ia. In Section 2, we will describe the data we will use and our method of calibration. To avoid any assumption on cosmological models, we will use the distance moduli of 580 SNe Ia from the Union2.1 sample to calibrate five GRB luminosity relations in the redshift range of SNe Ia sample . Then, the distance moduli of 82 high-z GRBs can be obtained from the five calibrated GRB luminosity relations. The joint constraints on the Cardassian expansion model and dark energy with 580 SNe and 82 calibrated GRBs’ data whose will be presented in Section 3. Finally, we will summarize our findings and present a brief discussion.
2. Calibrating the Updated Luminosity Relations of GRBs
2.1. Observational Data and Methodology
As mentioned above, we calibrate the updated luminosity relations of GRBs using low-z events whose distance moduli can be obtained by those of Type Ia supernovae. Actually, we use the cosmology-independent calibration method developed by [33–35]. This method is one of the interpolation procedures which use the abundant SNe Ia sample to interpolate the distance moduli of GRBs in the redshift range of SNe Ia sample (). More recently, the Supernova Cosmology Project collaboration released their latest SNe Ia dataset known as the Union2.1 sample, which contains 580 SNe detections . Obviously, there are rich SNe Ia data points, and we can make a better interpolation by using this dataset.
Our updated GRB sample includes 139 GRBs with redshift measurements; there are 57 GRBs at and 82 GRBs at . This sample is shown in Table 1, which includes the following information for each GRB: (1) its name; (2) the redshift; (3) the bolometric peak flux ; (4) the bolometric fluence ; (5) the beaming factor ; (6) the time lag ; (7) the spectral peak energy ; and (8) the minimum rise time . All of these data were obtained from previously published studies. Before GRB 060607, we take all the data directly from . We adopt the data between GRB 060707 and GRB 080721 from . For those GRBs detected after July 7, 2008, we adopt the data directly from . Applying the interpolation method, we can derive the distance moduli of 57 low-z GRBs and calibrate five GRB luminosity relations with this low-z sample, that is, the relation, the relation, the relation, the relation, and the relation. The isotropic peak luminosity of a burst is calculated by the isotropic equivalent gamma-ray energy is given by and the collimation-corrected energy is Here, is the luminosity distance of the burst, and are the bolometric peak flux and fluence of gamma-rays, respectively, while is the beaming factor, and is the jet half-opening angle. We assume each GRB has bipolar jets, and is the true energy of the bipolar jets.